Problem 4

Question

For Problems $1-12$, compute the indicated matrix by using the following matrices: $\begin{array}{ll} A=\left[\begin{array}{rr} 1 & -2 \\ 3 & 4 \end{array}\right] & B=\left[\begin{array}{lr} 2 & -3 \\ 5 & -1 \end{array}\right] \\ C=\left[\begin{array}{rr} 0 & 6 \\ -4 & 2 \end{array}\right] & D=\left[\begin{array}{rr} -2 & 3 \\ 5 & -4 \end{array}\right] \\ E=\left[\begin{array}{lr} 2 & 5 \\ 7 & 3 \end{array}\right] & \end{array}$ $$ 2 D-E $$

Step-by-Step Solution

Verified

Answer

The resulting matrix is $\begin{bmatrix} -6 & 1 \\ 3 & -11 \end{bmatrix}$.

1Step 1: Multiply Matrix D by 2

Given matrix $ D = \begin{bmatrix} -2 & 3 \ 5 & -4 \end{bmatrix} $, we need to multiply each element by 2: 1. First element: $-2 \times 2 = -4$ 2. Second element: $3 \times 2 = 6$ 3. Third element: $5 \times 2 = 10$ 4. Fourth element: $-4 \times 2 = -8$ So, $ 2D = \begin{bmatrix} -4 & 6 \ 10 & -8 \end{bmatrix} $.

2Step 2: Write Down Matrix E

Matrix $ E $ is given by: \[ E = \begin{bmatrix} 2 & 5 \ 7 & 3 \end{bmatrix} \]. We will use this matrix in the next step to perform the subtraction.

3Step 3: Subtract Matrix E from 2D

We perform the matrix subtraction: $ 2D - E = \begin{bmatrix} -4 & 6 \ 10 & -8 \end{bmatrix} - \begin{bmatrix} 2 & 5 \ 7 & 3 \end{bmatrix} $. The elements of the resulting matrix are calculated as follows: 1. $-4 - 2 = -6$ 2. $6 - 5 = 1$ 3. $10 - 7 = 3$ 4. $-8 - 3 = -11$ Therefore, the result is: \[ \begin{bmatrix} -6 & 1 \ 3 & -11 \end{bmatrix} \].

Key Concepts

Matrix SubtractionMatrix MultiplicationMatrix Addition

Matrix Subtraction

Matrix subtraction is the process of subtracting one matrix from another. It involves element-wise subtraction and requires both matrices to have the same dimensions.

First, check that the matrices have the same number of rows and columns. Incompatible sizes mean the subtraction cannot be performed.
Then, subtract the corresponding elements from each matrix.

Consider matrices 2D and E as in the example you have. Both matrices are 2x2, so subtraction can be performed:- Subtract each element in E from the corresponding element in 2D:

$-4 - 2 = -6$
$6 - 5 = 1$
$10 - 7 = 3$
$-8 - 3 = -11$

Matrix Multiplication

Matrix multiplication is different from addition and subtraction and requires a specific process. It's not just multiplying corresponding elements.

Ensure that the number of columns in the first matrix equals the number of rows in the second matrix.
Multiply each element of the row from the first matrix by each element of the column from the second matrix and sum the products.

For example, if matrix A is a 2x3 matrix and matrix B is a 3x2 matrix, you can multiply them to get a 2x2 matrix:- This involves taking the dot product of rows of A with columns of B.- Each resulting element in the new matrix is the sum of these products, such as in the following expression for one element:$(a_{11} \times b_{11} + a_{12} \times b_{21} + a_{13} \times b_{31})$Remember, the order in matrix multiplication matters, meaning $A \times B$ does not necessarily equal $B \times A$. Also, the size of the resulting matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix.

Matrix Addition

Matrix addition is one of the simpler operations involving matrices, as it only requires adding corresponding elements of the matrices.

First, ensure the matrices have the same dimensions, meaning the same number of rows and columns.
Add the elements in each position from both matrices.

For instance, given matrices A and B, to find the sum:- The element at the first row and first column in the resulting matrix is $a_{11} + b_{11}$.- This process continues for each element in the matrices:

$a_{12} + b_{12}$

$a_{21} + b_{21}$

$a_{22} + b_{22}$

The final matrix will have the same dimensions as the original matrices. If any element's addition requires further explanation, remember it is just an addition of two numbers, simplifying the problem significantly.

Problem 4

Other exercises in this chapter

Problem 4

For Problems $1-8$, find $A+B, A-B, 2 A+3 B$, and $4 A-2 B$. $$ A=\left[\begin{array}{r} 3 \\ -9 \\ 7 \end{array}\right], \quad B=\left[\begin{array}{r} -

View solution

Problem 4

For Problems $1-18$, find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rr} 2 & 9 \\ 3 & 13 \end{array}\right] $$

View solution

Problem 4

For Problems $1-24$, indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{arra

View solution

Problem 5

For Problems $1-8$, find $A+B, A-B, 2 A+3 B$, and $4 A-2 B$. $$ A=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -1 & 4 & -7 \\ 0 & 5 & 9 \end{array}\right], \qua

View solution