Use a computer or graphing calculator and the following...

Use a computer or graphing calculator and the following matrices to find the matrix products and sums. $$ A=\left[\begin{array}{rrrrr} 2 & 3 & -1 & 5 & 10 \\ 2 & 8 & 7 & 4 & 3 \\ -1 & -4 & -12 & 6 & 8 \\ 2 & 5 & 7 & 1 & 4 \end{array}\right] $$ $$ B=\left[\begin{array}{rrrr} 9 & 3 & 7 & -6 \\ -1 & 0 & 4 & 2 \\ -10 & -7 & 6 & 9 \\ 8 & 4 & 2 & -1 \\ 2 & -5 & 3 & 7 \end{array}\right] $$ $$ C=\left[\begin{array}{rrrrr} -6 & 8 & 2 & 4 & -3 \\ 1 & 9 & 7 & -12 & 5 \\ 15 & 2 & -8 & 10 & 11 \\ 4 & 7 & 9 & 6 & -2 \\ 1 & 3 & 8 & 23 & 4 \end{array}\right] $$ $$ D=\left[\begin{array}{rrrrr} 5 & -3 & 7 & 9 & 2 \\ 6 & 8 & -5 & 2 & 1 \\ 3 & 7 & -4 & 2 & 11 \\ 5 & -3 & 9 & 4 & -1 \\ 0 & 3 & 2 & 5 & 1 \end{array}\right] $$ a. Find $C+D$. b. Find $(C+D) B$. c. Find $C B$. d. Find $D B$. e. Find $C B+D B$. f. Does $(C+D) B=C B+D B ?$

Use a computer or graphing calculator and the following matrices to find the matrix products and sums.
$$
A=\left[\begin{array}{rrrrr}
2 & 3 & -1 & 5 & 10 \\
2 & 8 & 7 & 4 & 3 \\
-1 & -4 & -12 & 6 & 8 \\
2 & 5 & 7 & 1 & 4
\end{array}\right]
$$
$$
B=\left[\begin{array}{rrrr}
9 & 3 & 7 & -6 \\
-1 & 0 & 4 & 2 \\
-10 & -7 & 6 & 9 \\
8 & 4 & 2 & -1 \\
2 & -5 & 3 & 7
\end{array}\right]
$$
$$
C=\left[\begin{array}{rrrrr}
-6 & 8 & 2 & 4 & -3 \\
1 & 9 & 7 & -12 & 5 \\
15 & 2 & -8 & 10 & 11 \\
4 & 7 & 9 & 6 & -2 \\
1 & 3 & 8 & 23 & 4
\end{array}\right]
$$
$$
D=\left[\begin{array}{rrrrr}
5 & -3 & 7 & 9 & 2 \\
6 & 8 & -5 & 2 & 1 \\
3 & 7 & -4 & 2 & 11 \\
5 & -3 & 9 & 4 & -1 \\
0 & 3 & 2 & 5 & 1
\end{array}\right]
$$

a. Find $C+D$.
b. Find $(C+D) B$.
c. Find $C B$.
d. Find $D B$.
e. Find $C B+D B$.
f. Does $(C+D) B=C B+D B ?$

Key Concepts

Dimension Compatibility

Dimension compatibility is a fundamental concept in matrix operations which ensures that matrices have the correct dimensions for operations such as addition or multiplication. For addition, both matrices must be of the same size, while for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

Distributive Property of Matrix Multiplication Over Addition

The distributive property in matrix algebra states that the multiplication of a matrix by the sum of two matrices is equivalent to the sum of the products of the matrix with each addend individually. This property allows one to simplify expressions and confirm the equivalence between (C+D)*B and C*B + D*B in matrix computations.

Matrix Addition

Matrix addition is the operation of adding two matrices of the same dimensions by adding the corresponding entries in each matrix. The result is a matrix of the same dimensionality, and this operation is both commutative and associative, meaning the order in which matrices are added does not affect the sum.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix, where each entry in the product is computed as the sum of the products of corresponding entries from the rows of the first matrix and the columns of the second matrix. This operation is defined only when the number of columns in the first matrix equals the number of rows in the second matrix.

Transcript

00:01 For this problem, we have been given four matrices, a, b, c, and d.

00:06 And we can use either a computer or a graphic calculator to do some matrix arithmetic.

00:12 For this problem, i'm using the desmos graphing application on my computer.

00:17 And as you can see, i have already entered in a, b, c, and d into this application.

00:23 So now we're just going to be doing some various arithmetic problems.

00:28 Before we do, though, let's take a look at what we want to find in this problem.

00:34 If we were using just numbers or variables, not matrices, if i gave you a problem like this, c plus d in parentheses times b equals cb plus db, we would say, of course, this is the distributive property.

00:54 We can distribute multiplication over addition.

00:57 But what we want to find out here is, does this property, does this distributive property also apply to matrices? so we're going to find a series of additions, you know, sums and products.

01:14 We'll find c plus d, and then we will multiply that by b, so that'll show us the one side.

01:21 And then we will find both of these products, c, b and c, d, and then add them together.

01:28 And we're going to see if what we get on the right -hand side of this equation does indeed or does not match the left -hand side.

01:37 Let's get started with this.

01:39 First, c -plus -d.

01:42 C -plus -d.

01:45 So that gives us a 5 -5 matrix, which makes sense.

01:50 When you add matrices, they need to be the same size.

01:53 And if i come up, you'll see that c and d are both 5 -5.

01:56 So it results in a 5 -5 matrix.

01:59 Now, let's take c plus d, and i'm going to multiply that by matrix b.

02:09 So i have a 5x4 matrix.