Question

Find (a) \(A+B\), (b) \(A-B\), (c) \(3 A\), and (d) \(3 A-2 B\). $$ A=\left[\begin{array}{rr} 7 & 4 \\ -4 & 5 \end{array}\right], B=\left[\begin{array}{rr} -3 & 1 \\ 8 & -4 \end{array}\right] $$

Short answer

The answers are: (a) \( A + B = \left[\begin{array}{cc} 4 & 5 \ 4 & 1 \end{array}\right] \), (b) \( A - B = \left[\begin{array}{cc} 10 & 3 \ -12 & 9 \end{array}\right] \), (c) \( 3A = \left[\begin{array}{cc} 21 & 12 \ -12 & 15 \end{array}\right] \), and (d) \( 3A-2B = \left[\begin{array}{cc} 27 & 10 \ -32 & 23 \end{array}\right] \).

Step-by-step solution

Matrix Addition (A + B)

To add two matrices, add the corresponding components from each matrix. Thus, the result of adding matrices A and B yields a new matrix \[C = A + B = \left[\begin{array}{cc} 7+(-3) & 4+1 \ -4+8 & 5+(-4) \end{array} \right] = \left[\begin{array}{cc} 4 & 5 \ 4 & 1 \end{array} \right]\]

Matrix Subtraction (A - B)

Subtraction is conducted similarly to matrix addition by subtracting the corresponding components. Hence, the subtraction of matrix B from A results in a new matrix \[D = A - B = \left[\begin{array}{cc} 7-(-3) & 4-1 \ -4 - 8 & 5 - (-4) \end{array} \right] = \left[\begin{array}{cc} 10 & 3 \ -12 & 9 \end{array} \right]\]

Scalar Multiplication (3A)

For this computation, the scalar number is multiplied with each entry in the matrix. Thus, the result of multiplying 3 with every entry in matrix A yields a new matrix \[E = 3 \cdot A = \left[\begin{array}{cc} 3 \cdot 7 & 3 \cdot 4 \ 3 \cdot (-4) & 3 \cdot 5 \end{array} \right] = \left[\begin{array}{cc} 21 & 12 \ -12 & 15 \end{array} \right]\]

Combination of Operations (3A - 2B)

This requires both scalar multiplication and matrix subtraction. First, perform the scalar multiplication for each matrix. Afterward, subtract the results. Thus, \[F = 3A - 2B = \left[\begin{array}{cc} 3 \cdot 7 - 2 \cdot (-3) & 3 \cdot 4 - 2 \cdot 1 \ 3 \cdot (-4) - 2 \cdot 8 & 3 \cdot 5 - 2 \cdot (-4) \end{array} \right] = \left[\begin{array}{cc} 27 & 10 \ -32 & 23 \end{array} \right]\]

Key concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. It's a simple yet fundamental operation in linear algebra that requires each matrix to have the same dimensions. Matrices of different sizes cannot be added because there would be unequal numbers of elements in each matrix.

To add two matrices, simply add the corresponding elements. For example, if matrix A is \( \begin{bmatrix} 7 & 4 \ -4 & 5 \end{bmatrix} \) and matrix B is \( \begin{bmatrix} -3 & 1 \ 8 & -4 \end{bmatrix} \), place them in the same locations in a new matrix:

• Top left: \( 7 + (-3) = 4 \)
• Top right: \( 4 + 1 = 5 \)
• Bottom left: \( -4 + 8 = 4 \)
• Bottom right: \( 5 + (-4) = 1 \)

Thus, the result of the addition, A + B, is the matrix \( \begin{bmatrix} 4 & 5 \ 4 & 1 \end{bmatrix} \). This process is very straightforward once you ensure the matrices have the same size.

Matrix Subtraction

Matrix subtraction follows the same principles as matrix addition, namely that both matrices must be of the same dimensions. The goal is to subtract the elements of one matrix from the corresponding elements of the other matrix.

For example, to find the result of A - B, where A and B are the given matrices \( \begin{bmatrix} 7 & 4 \ -4 & 5 \end{bmatrix} \) and \( \begin{bmatrix} -3 & 1 \ 8 & -4 \end{bmatrix} \) respectively, follow these steps:

• Top left: \( 7 - (-3) = 10 \)
• Top right: \( 4 - 1 = 3 \)
• Bottom left: \( -4 - 8 = -12 \)
• Bottom right: \( 5 - (-4) = 9 \)

Thus, matrix A minus matrix B gives us the resulting matrix \( \begin{bmatrix} 10 & 3 \ -12 & 9 \end{bmatrix} \). As with matrix addition, make sure the matrices are equally sized before performing subtraction, otherwise, it's not possible.

Scalar Multiplication

Scalar multiplication is an operation where each entry of a matrix is multiplied by a single constant, called the scalar. Unlike addition and subtraction, this operation doesn't require another matrix. Instead, it involves only one matrix and one scalar.

Consider the matrix A\( \begin{bmatrix} 7 & 4 \ -4 & 5 \end{bmatrix} \) multiplied by the scalar 3. Multiply every element of A by 3:

• Top left: \( 3 \times 7 = 21 \)
• Top right: \( 3 \times 4 = 12 \)
• Bottom left: \( 3 \times (-4) = -12 \)
• Bottom right: \( 3 \times 5 = 15 \)

The resulting matrix after scalar multiplication is \( \begin{bmatrix} 21 & 12 \ -12 & 15 \end{bmatrix} \). Scalar multiplication scales the matrix by distributing the scalar across all elements, making it a very versatile and powerful operation for transforming matrices.