Matrices: The Basics

Hello and welcome to this video about the basics of matrices! In this video, we will cover:

• What matrices are
• Some components of matrices
• Some types of matrices
• Matrix operations

A matrix is commonly defined as a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices can be used in several different algebraic and geometric situations.
Here’s the general form of a matrix…

As you can see, it is made of an undetermined number of rows and columns.

An example of a matrix might look like this:

Matrices are usually written in square brackets and they are commonly classified by their dimension. Matrix A is a 3×2 matrix because it consists of 3 rows and 2 columns. Each of the numbers in matrix A [A] is called an element or an entry. Each element can be classified according to its position within [A]. \(A_{m,n}\) designates the element at row m column n. For example, \(A_{2,1}=6\) because 6 is the element located at row 2 column 1 of the matrix.

Some matrices consist of only one row or column. These are known as vectors.

Matrix B might also be classified as a row matrix and Matrix C might also be classified as a column matrix. Notice that Matrix B is a 1×3 matrix, while Matrix C is a 3×1 matrix.

The elements where m=n constitute the main diagonal of a matrix. In this matrix here,

the elements \(A_{1,1}\) and \(A_{2,2}\) are on the main diagonal. So the entries negative 1 and negative 3 constitute the main diagonal. In matrices with unequal numbers of rows and columns, the main diagonal isn’t nearly as useful as it is with square matrices – matrices with an equal number of rows and columns. Here are some examples of square matrices:

Matrix D might be called a lower triangular matrix, because all the entries above the main diagonal are 0.

Matrix E might be called an upper triangular matrix because all the entries below the main diagonal are 0.
Note that triangular matrices can have entries that are 0 on the main diagonal, like [E] does.

Matrix F has the qualities of both [D] and [E]. The entries are 0 everywhere but the main diagonal. Instead of calling this matrix triangular, we simply call it diagonal.

Diagonal matrices whose entries on the main diagonal are all 1s are called identity matrices. Matrix G is known as the 2 by 2 identity matrix, notated by I sub 2.\(I_{2}\).

Sometimes, matrices like [H] have entries that are all 0s. [H] is known as the 2×2 zero matrix or null matrix, which is notated as \(0_{2×2}\).

In order for matrices to be equal, they must contain the same elements in the same positions.

In this case, [I]=[K], but [I] ≠[J] and [J] ≠ [K].

Matrices can be added and subtracted if they have the same dimension. To add or subtract matrices, simply add or subtract the corresponding elements.

[D] is a 3×3 matrix, as is [E], so they can be added or subtracted. Matrix D plus Matrix E (you’re just going to add the corresponding components) so 2 plus 0, 6 plus 0, negative 9 plus 0, 0 plus 99, 7 plus 1, 12 plus 0, 0 plus 2, 0 minus 15, 14 plus 4; and then from there you just simplify. So we have 2, 6, negative 9, 99, 8, 12, 2, negative 15, and 18. For subtraction you’ll do the same steps but with the different operation. This time let’s do [E] minus [D]. Remember, when adding or subtracting matrices they must be the same size. And then all you do is add or subtract the matching components for each matrix.

Matrices can be multiplied by a scalar. To multiply a matrix by a scalar, simply multiply each element of the matrix by the scalar. So what we’re going to do is multiply 3 by each of these different elements. So we have 3 times 2, 3 times 6, 3 times negative 9, 3 times 0, 3 times 7, 3 times 12, 3 times 0, 3 times 0, 3 times 14. Which then gives us 6, 18, negative 27, 0, 21, 36, 0, 0, and 42. And it’s as easy as that! Simply take your matrix and multiply by the scalar, which in this case is 3.

In some cases, matrices can be multiplied by each other. The dimensions of the matrices tell us two things: (1) whether multiplication is possible, and (2) the dimension of the product.

In order for matrix multiplication to be possible, the inner dimensions must match. In other words, the number of columns of the left matrix must match the number of rows of the right matrix. Suppose we wanted to multiply [A] times [D]. [A] is a 3×2 matrix and [D] is a 3×3 matrix.

3× 2∙3 ×3

Since the inner dimensions don’t match, we cannot multiply [A][D]. However, we can multiply [D] times [A]. You’ll get a 3×3 matrix times a 3×2 matrix.

3× 3∙3 ×2

The number of columns of [D] matches the number of rows of [A]. Furthermore, the outer dimensions specify the dimensions of the product.

3× 3∙3 ×2

The dimension of [D]times [A] will be a 3×2 matrix. Before we actually do the multiplication, it’s important to notice a major concept here. With numbers, multiplication commutes. 3×4 = 4×3 = 12. With matrices, multiplication does not commute. The order of the matrices matters. As we just saw, [A][D]≠ [D][A].
Here’s how matrix multiplication works:

The pattern to notice here is that we move across the rows of Matrix D and down the columns of Matrix A. So we multiply element 1 of Matrix D row 1 times element 1 of Matrix A column 1, then add element 2 of [D] row 1 multiplied by element 2 of [A] column 1 then add element 3 of [D] row 1 multiplied by element 3 of [A] column 1. The result of row 1 of [D] times column 1 of [A] becomes row 1, column 1 of the product.

Now here are a few other properties to make note of. We’ll use [D] to illustrate:

• \(0_{3×3}+D=[D]\) The zero matrix acts like the number 0 when adding numbers – the additive identity. A number plus 0 always equals itself.
• \(0{3×3} \times D=D \times 0_{3×3}=0_{3×3}\) The zero matrix acts like the number 0 when multiplying numbers. A number times 0 equals 0.
• \([D]· I_{3} = I_{3}·[D] = [D]\). The identity matrix acts like the number 1 does in multiplication of numbers – it’s the multiplicative identity. A number times 1 equals itself.

Thanks for watching, and happy studying!

Matrices - Practice Questions