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INSTRUCTORSCarleen EatonGrant Fraser
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Lecture Comments (6)

1 answer

Last reply by: Rita Semaan
Mon Jan 7, 2013 8:10 PM

Post by Rita Semaan on January 7, 2013

For example one, when you were finding the first row and column you multiplied zero by zero then one by zero and added then said it was one. Shouldn't it be zero or did you multiply the one but the one on the second column?

3 answers

Last reply by: Dr Carleen Eaton
Tue Nov 29, 2011 8:56 PM

Post by Michael Fabrikant on January 22, 2011

In example three, why did the instructor multiply the bottom row by 1/3 and not negative 1/3?

Identity and Inverse Matrices

  • The identity matrix plays the role of the identity under multiplication.
  • The inverse of a matrix represents the inverse under multiplication.
  • A square matrix has an inverse if and only if its determinant is not zero.
  • The inverse of a 2 x 2 determinant can be calculated using a specific formula.

Identity and Inverse Matrices

Find the inverse of A if it exist [
− 10
4
5
− 1
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 10( − 1) − (4)(5))][
    − 1
    − 4
    − 5
    − 10
    ] =
  • − [1/10][
    − 1
    − 4
    − 5
    − 10
    ] = [
    [1/10]
    [2/5]
    [1/2]
    1
    ]
[
[1/10]
[2/5]
[1/2]
1
]
Find the inverse of A if it exist [
3
0
3
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((3)(0) − (0)(3))][
    0
    0
    − 3
    3
    ] = [1/0] [
    − 1
    − 4
    − 5
    − 10
    ] = Inverse does not exist. You cannot divide by zero.
Inverse does not exist. You cannot divide by zero.
Find the inverse of A if it exist [
5
− 1
8
− 1
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((5)( − 1) − ( − 1)(8))][
    − 1
    1
    − 8
    5
    ] =
  • [1/3][
    − 1
    1
    − 8
    5
    ] = [
    − [1/3]
    [1/3]
    − [8/3]
    [1/3]
    ]
[
− [1/3]
[1/3]
− [8/3]
[1/3]
]
Find the inverse of A if it exist [
− 5
6
4
− 4
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 5)( − 4) − (6)(4))][
    − 4
    − 6
    − 4
    − 5
    ] = − [1/4][
    − 4
    − 6
    − 4
    − 5
    ] = [
    1
    [3/2]
    1
    [5/4]
    ]
[
1
[3/2]
1
[5/4]
]
Find the inverse of A if it exist [
− 1
0
− 2
− 3
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)( − 3) − (0)( − 2))][
    − 3
    0
    2
    − 1
    ] = [1/3][
    − 3
    0
    2
    − 1
    ] = [
    − 1
    0
    [2/3]
    − [1/3]
    ]
[
− 1
0
[2/3]
− [1/3]
]
Find the inverse of A if it exist [
− 1
− 8
1
8
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)(8) − ( − 8)(1))][
    8
    8
    − 1
    − 1
    ] = [1/0][
    8
    8
    − 1
    − 1
    ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
− 1
− 2
3
10
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)(10) − ( − 2)(3))][
    10
    2
    − 3
    − 1
    ] = −[1/4] [
    10
    2
    3
    − 1
    ]
−[1/4] [
10
2
3
− 1
]
Find the inverse of A if it exist [
5
0
− 6
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((5)(0) − (0)( − 6))][
    0
    0
    6
    5
    ] = [1/0] [
    0
    0
    6
    5
    ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
− 3
− 3
3
3
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b − c
    a
    ] = [1/(( − 3)(3) − ( − 3)(3))][
    3
    3 − 3
    − 3
    ] = [1/0][
    3
    3
    − 3
    − 3
    y ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
8
− 5
− 1
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((8)(0) − ( − 5)( − 1))][
    0
    5
    1
    8
    ] = −[1/5][
    0
    5
    1
    8
    ]
−[1/5][
0
5
1
8
]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Identity and Inverse Matrices

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Identity Matrix 0:13
    • Example: 2x2 Identity Matrix
    • Example: 4x4 Identity Matrix
    • Properties of Identity Matrices
    • Example: Multiplying Identity Matrix
  • Matrix Inverses 5:30
    • Writing Matrix Inverse
  • Inverse of a 2x2 Matrix 6:39
    • Example: 2x2 Matrix
  • Example 1: Inverse Matrix 10:18
  • Example 2: Find the Inverse Matrix 13:04
  • Example 3: Find the Inverse Matrix 17:53
  • Example 4: Find the Inverse Matrix 20:44

Transcription: Identity and Inverse Matrices

Welcome to Educator.com.0000

In today's lesson, we are going to continue on talking about matrices, this time focusing on two special types of matrices, identity and inverse matrices.0002

The identity matrix is a square nxn matrix, which has 1 for every element in the main diagonal, and 0 for every other element.0013

So, an identity matrix is a square matrix...and let's look at an example--for example, a 2x2 matrix that is an identity matrix.0026

It says that this has 1 for every element in the main diagonal; so that is 1 and 1; and 0 for every other element.0036

Looking at another example of this for a 4x4 matrix: in the main diagonal (that would be here), I have all 1's.0048

And I am going to fill in the other spots with 0's; so, down the main diagonal, I have 1's; everything else is a 0.0063

And this is 1, 2, 3, 4; 1, 2, 3, 4; so it is a 4x4 square matrix.0077

Now, let's talk about properties of these matrices.0083

For any nxn matrix A, if you multiply that matrix times its identity matrix, then you will get the original matrix back.0087

And you can also multiply these in either order.0097

So, in a way, when you think about identity--the identity property of multiplication--think back to regular numbers;0101

and when we talked about identity with multiplication, we would say that, if you multiplied the number n, any number,0112

times 1, you got that original number back; and that was the identity property with multiplication.0123

For example, if I took 3 times 1, I am going to get 3; or 10 times 1--I will get 10 back.0130

This is the same idea, but with matrices; so, this functions the same way as 1 in this case,0136

because if you multiply the matrix times its identity matrix, then you will get the original matrix back--0143

the same way as, if you multiplied a number times 1, you get the number back.0153

So, let's go ahead and try this--and remember that, unlike 1, though...it is just the number 1; but for identity matrices, there are multiple 1's.0157

You see that, for a 2x2 matrix, the identity matrix will be different than for a 4x4 or a 3x3.0164

OK, so let's use this 2x2, and let's say I had some matrix A, and it is going to be 3, -1, 2, 5.0170

And I am going to multiply it by its identity matrix, which is going to be this one, because it's a 2x2; and let's see what I get.0184

Well, recall that, for multiplication, if I am going to look for this position, row 1, column 1,0194

I am going to multiply row 1 of this first matrix times this first column here; so that is going to give me 3 times 1.0206

And then, I am going to find the sum of those products: 3 times 1, plus -1 times 0.0215

That is going to give me 3 plus 0, or 3; OK, so I am going to get 3 up here.0224

And let's look for row 1, column 2; it will be row 1, column 2; so 3 times 0, plus -1 times 1.0237

This is going to give me 0 - 1; 0 - 1 is -1, so -1 goes right here.0251

And continuing on now to the second row: row 2, column 1: row 2 of A, times column 1 of the identity matrix,0262

is going to give me 2 times 1, plus 5 times 0, which is going to equal...2 + 0 is 2.0274

OK, then finally, row 2, column 2: this is going to give me row 2, column 2--row 2 here, column 2 here.0287

Row 2 is 2 times 0, plus 5 times 1; well, that is simply 0 + 5, or 5.0297

So, you see that this property is shown here--that if I have a matrix A, and I multiply it by its identity matrix I, I get A back.0306

In the same way, for numbers, with the identity property of multiplication: if I take a number and multiply it by 1, I get the original number back.0317

OK, now talking about matrix inverses: if we have two nxn matrices (these are square matrices),0328

and we say that they are inverses of each other...if I have two nxn matrices, A and B, they are inverses of each other0339

if AB is equal to BA, and that product AB is equal to the identity matrix.0345

So, if the product of A and B turns out to be the identity matrix, then those two matrices are inverses of each other.0353

And the inverse of A, if it exists (and we are going to talk about that in a few minutes--that it may not always exist, and why),0368

is written as A-1: we pronounce this "A inverse."0374

OK, again, let's say I was given two matrices, A and B, and then I was asked, "Are they inverses?"0378

The way I would determine that is by multiplying those two out; and if the product is the identity matrix, then they are inverses.0389

If it is not, then those two are not inverses of each other.0395

Let's talk about finding the inverse of a 2x2 matrix.0399

If we have some matrix, A, that looks like this; the inverse of A, if it exists, is given by this formula: A-1 = 1/...0404

and if you look at this, ad - bc, this is going to look familiar; and that is the determinant of this matrix.0414

Remember that the determinant of a 2x2 matrix is given by the formula ad - bc; so it is 1 over the determinant, times this matrix.0423

And it is pretty easy to remember how you form this matrix, if you just look at it this way.0432

The way I got this is: I switched the positions of a and d, and then I took the opposite sign of b and the opposite sign of c.0438

So, if I had numbers here...let's look at that: if I was told that A is the matrix 3, 4, 1, 2, and I was told to find A-1:0449

well, then I would take 1 over the determinant, which is 3(2) - 4(1), times...0466

now, to find this matrix, I am going to switch these positions; I am going to put 2 here and 3 there.0481

I am going to take this number; and I keep the number, but I am going to put the opposite sign.0486

If this was a -4, I would have made it for; the same for the c position--I am going to keep it 1, but I am going to make it the opposite sign.0492

Now, stopping here for a second: I mentioned "if the inverse exists."0500

Looking at this, you can see that there could be a situation where the inverse does not exist.0505

OK, here I have 1/(6 -4); so that is 1/2; well, that is allowable.0511

But if this had been, say, 6 - 6, that would have been 0; that is not allowable--that is undefined.0518

So, I wouldn't have been able to find A-1.0524

OK, therefore, there are situations (and it is when this denominator turns out to be 0) that you can't find the inverse.0527

So, it is a good idea to look for this denominator right away, before you do any more work,0534

to determine if we even can find the inverse, so you don't waste any more time looking for something that does not exist.0539

Now that I have found that 1 over the determinant is actually 1/2,0546

I can just multiply using scalar multiplication (this is functioning as a scalar) to figure out A1.0552

So, I am just going to multiply 1/2 times each element: 1/2 times 2 is 2/2, which is 1.0562

And 1/2 times -4 is -4/2; that is -2; 1/2 times -1 is -1/2; and then, 1/2 times 3 is 3/2.0570

Now, if you wanted to check your work, you could always check your work by taking A times A-1,0584

and seeing if it comes out to the identity matrix, which it should if you did things correctly.0593

If I were to take 3, 4 times 1, 2, and multiply it by 1, -2, -1/2, 3/2, I would find that it does come out to the identity matrix for these 2x2 matrices.0598

OK, so first we are asked, in Example 1, to determine if A and B are inverses of each other.0617

So, are A and B inverses of each other? We are given these two matrices and asked if they are inverses.0625

Well, recall that if two matrices are inverses of each other, then if I multiply them,0631

I can actually multiply them in either order, and I am going to get the identity matrix.0637

So, before I proceed, let's just think about what the identity matrix is for a 2x2.0645

And I am going to have 1 along the main diagonal and 0 everywhere else, so this is the identity matrix in this situation.0649

I am going to go ahead and multiply these and see what I end up with.0654

OK, I am doing some matrix multiplication: first, row 1, column 1--first row, first column--that is 0 times 0, plus 1 times 1.0663

And that is going to give me 1 in this position.0684

Now, row 1, column 2: 0 times 0, plus...OK, then that would be 1, 1, times 2...row 1, column 2: I did 0 times 0, and now 1 times 2.0687

That is 0 plus 2, so that is 2; so in this position, I am going to get 2.0710

Now, row 2, column 1: 2 times 0, plus 1 times 1--that is going to give me 0 + 1 = 1 right here.0715

Row 2, column 2: 2 times 0, plus 1 times 2: 0 plus 2 is 2.0734

All right, so the question I was asked is if these are inverses of each other.0750

Well, when I took their product (the product of A and B), I found that AB does not equal the identity matrix.0755

This is not the identity matrix; this is; therefore, are A and B inverses of each other?0765

No, A and B are not inverses of each other.0770

And I was able to check that using this property.0781

Find the inverse if it exists: so finding the inverse of a 2x2 matrix, recall, uses this formula: 1 over the determinant,0786

which is ad - bc, times the matrix d, -b, -c, a.0796

A-1 (we are calling this matrix A) is 1 over the determinant.0809

Here I have 1 times 4, minus 2 times 3.0815

And just looking at this quickly, I see that this is 4 - 6, so that is -2; and therefore, that inverse is -1/2, so this inverse exists.0820

I didn't get 0 down here (before I proceed any farther).0832

That, times...da means I am going to switch these two: 4 will go in the a position; 1 will go in the d position.0836

For b, I am just going to take the opposite sign; and for c, I am just going to take the opposite sign.0845

Proceeding: this is going to give me 4 minus 6, as I said, which is going to give me -1/2, times 4, -2, -3, 1.0852

Now, I need to multiply each element in here by -1/2; and that is going to give me -1/2, times 4, which is -2;0862

-1/2 times -2 is -2/-2, which is 1; -1/2 times -3 is going to give me positive 3/2; and -1/2 times 1 is -1/2.0874

Now, if I wanted to check this, I could check it by saying that A times A-1 is the identity matrix.0893

This is A-1: let's go ahead and multiply this times A and see what happens.0900

So, let's try A times A-1, and just see what we get.0907

Here, I had A; that is 1, 2, 3, 4; and A-1, and that is -2, 1, 3/2, -1/2, equals...0913

OK, working over here, row 1, column 1: that is 1, and I am just going to go ahead and do part of this mentally...0937

1 times -2 is -2; 2 times 3/2...well, this cancels out; that just gives me 3.0951

This is going to give me -2 + 3 = 1; so I get 1 right here.0960

OK, row 1, column 2: that is 1 times 1 is 1; 2 times -1/2 is -2/2, which is -1.0965

Since 1 - 1 is 0, 0 goes in the row 1, column 2 position.0983

Let's go on to row 2, column 1: it is going to give me 3 times -2; that is -6.0990

And then, this is 4 times 3/2 equals 12/2; that equals 6.1009

So, -6 + 6 = 0; so I get 0 right here.1015

Row 2, column 2--the last position--equals: 3 times 1 is 3; 4 times -1/2 would be -4/2, or -2; 3 minus 2 is 1.1024

So you see, I actually did get the identity matrix; so I figured out A-1, and I was checking1046

that I was correct by saying, "Well, if this truly is the inverse, when I multiply these two together--1051

if those two matrices are inverses of each other, I will get the identity matrix."1057

And this is the identity matrix for a 2x2 matrix.1061

So, the inverse--the answer to the question they are asking--is this right here.1065

However, I checked my work right there.1070

Example 3: Find the inverse, if it exists.1074

I am using my formula, A-1 = 1 over the determinant, ad - bc, times...switching d and a and changing the signs of b and c.1079

So, A-1 equals 1 over...this is ad; that is -1 times 0, minus bc (minus -3 times -1).1095

Then, times this matrix...it was found by switching these two positions, 0 and -1.1110

Now, for -3, which is in the b position, I am going to take the opposite sign; that is going to be 3.1116

For -1, which is in the c position, I am going to take the opposite sign and make it 1.1122

OK, let's make sure that this is going to work now.1127

This is 0 minus -3 times -1; that is going to give me 3, so that is 1 over 0 minus 3; and there will be an inverse, because this is not 0.1131

If this had come out to be 0, I couldn't have found the inverse.1150

It is -1/3; 1 over 0 - 3 is -1/3...times this matrix.1154

This is going to give me, if I multiply -1/3 by each element in here...I am going to get -1/3 times 0, and that is going to give me 0.1164

Here, I have -1/3 times 3; that is going to give me -3/3, or -1, right here.1176

1/3 times 1 is 1/3; and 1/3 times -1 is -1/3.1187

So again, I was asked to find the inverse; I used this formula.1194

I first found the determinant, and I determined that it was -1/3; therefore, I could find the inverse.1198

And then, I multiplied it by the matrix, which consists of switching the positions of these two and reversing the signs of these two.1207

This becomes a positive; this also becomes a positive--we are giving them the opposite signs from what they had.1218

Then, I multiplied -1/3 by each element in this matrix to get this.1224

And if I were to check it (which I could), it would be by multiplying A (the original matrix) times its inverse.1229

And I would find that I get the identity matrix back.1238

OK, find the inverse if it exists: again, I am recalling my formula: A-1 is 1 over the determinant, ad - bc,1241

times the matrix found by switching a and d, and reversing the sign on b, and reversing the sign on c.1259

OK, so first the determinant: that is 1 times 6, minus 2 times 3.1269

The matrix: switch these two positions, and reverse the sign on 3, and reverse the sign there.1279

Now, you probably already saw that I didn't even need to go that far, because there is a problem.1286

What I have here is 1 times 6 (which is 6), minus 6, times its matrix.1291

Well, this turns out, obviously, to be 0; and since you can't have that--that is not allowed--it is undefined--1301

we can stop right here, because this is undefined.1311

In this case, the inverse does not exist; and the clue is: as soon as you see1317

that you are having to divide by 0, you know that you are not allowed to do that.1329

So, the inverse does not exist.1333

That concludes this lesson on Educator.com on identity and inverse matrices, and I will see you again soon!1338