In this lesson, our instructor Vincent Selhorst-Jones teaches Matrices. Youll learn the definition of a matrix, about size and dimension, what a square matrix is, when capital letters are used, and when two matrices are equal. Vincent goes over some examples and then dives right into specific entries. Youll learn how to talk about matrices and scalar multiplication, as well as, how to add and multiply them. Vincent ends the lesson with slides on special matrices, such as the zero and identity matrix, and then finishes with four examples.
Note: Many teachers and textbooks first introduce matrices as a way to solve systems of linear equations through augmented matrices, row operations, and Gauss-Jordan elimination. If you're looking for that, watch the first
part of the lesson Using Matrices to Solve Systems of Linear Equations.
A matrix (plural: matrices) is a rectangular array where each entry is a number.
For a matrix with m rows and n columns, we say it has an order of m×n (This property is sometimes called `size' or `dimension'). We can also write order as Am ×n. If a matrix has equal numbers of rows
and columns (m=n), we call it a square matrix.
Matrices are usually denoted by capital letters.
Two matrices A and B are equal if they have the same order and all their entries are equal.
We can also talk about some specific entry in row i and column j (where i and j are standing in for numbers). As we use capitals to denote a matrix (A), we often use the corresponding lower case to denote its entries (a). We can talk about a specific
entry by using the subscript ij on the letter (aij) to denote the ith row and jth column.
With this idea in mind, we can see a matrix as a series of entries represented by various aij. This means instead of having to write the entire matrix out (like above) or just using a letter to denote the whole thing (A), we can refer to it
by using a single representative entry to stand in for all entries:
A = [ aij ].
Since i and j can change, aij is a placeholder for all of the entries in A.
Given some matrix A and a scalar (real number) k, we can multiply the matrix by the number:
kA = [ k ·aij ].
That is, every entry of A is multiplied by k. [Note that this is just like multiplying a vector by a scalar.]
Given two matrices A and B that have the same order (m×n, the number of rows and columns), we can add the two matrices together:
A + B = [aij + bij ].
That is, we add together entries that come from the same "location" in each matrix to create a new matrix. [Note that this is very similar to adding vectors component-wise.]
If A is an m×n matrix and B is an n ×p matrix, we can multiply them together and create a new matrix AB that is order m×p, and which is defined as
AB = [cij],
where cij = ai1b1j + ai2b2j + …+ ainbnj. That is, entry cij of AB (the entry in its ith row and jth column) is the sum of the products of corresponding
entries from A's ith row & B's jth column. [The idea of matrix multiplication can be very confusing at first. Check out the video to see a lot of visual references to help explain what's going on here.]
To multiply two matrices together, we have to first be sure that their orders are compatible. The numbers of columns in the first matrix must equal the number of rows in the second matrix.
Multiplication in the real numbers is commutative, that is, x·y = y ·x: which side you multiply from does not affect the product. ( 5·7 = 7 ·5, 8(−3) = (−3)8 ). However, matrix multiplication
is not commutative in general. That is, for most matrices A and B, AB ≠ BA.
The zero matrix is a matrix that has 0 for all of its entries. A zero matrix can be made with any order. It is denoted by 0. [If you need to show its order: 0m×n.]
The identity matrix is a square matrix that has 1 for all its entries on the main diagonal and 0 for all other entries. It can be any order, so long as it is square. It is denoted by I. (If you need to show it is
order n ×n, you can denote by: In.) Notice that for any matrix A, IA = A = A I. [ I effectively works the same as multiplying
a real number by 1.]
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.