Problem 71 - Algebra 2

Question

Use matrices \(A, B, C,\) and \(D\) to find the following. $$ A=\left[\begin{array}{rr}{-4} & {4} \\ {2} & {-3} \\ {1} & {5}\end{array}\right] \quad B=\left[\begin{array}{rr}{7} & {0} \\ {4} & {1} \\\ {6} & {-2}\end{array}\right] \quad C=\left[\begin{array}{rr}{-4} & {-5} \\\ {-3} & {1} \\ {2} & {3}\end{array}\right] \quad D=\left[\begin{array}{rr}{1} & {-2} \\ {1} & {-1} \\ {-3} & {4}\end{array}\right] $$ $$ 3 B-2 A $$

Step-by-Step Solution

Answer

The result of \(3B - 2A\) is \(\left[\begin{array}{rr} 29 & -8 \\ 8 & 9 \\ 16 & -16 \end{array}\right]\).

Step 1: Understand the Problem Statement

The problem asks us to calculate the matrix expression \(3B - 2A\). This involves scalar multiplication and matrix subtraction.

Step 2: Multiply Matrix B by 3

We will perform scalar multiplication on matrix \(B\). Multiply each element of matrix \(B\) by 3:\[3B = 3 \times \left[ \begin{array}{rr} 7 & 0 \ 4 & 1 \ 6 & -2 \end{array} \right] = \left[ \begin{array}{rr} 21 & 0 \ 12 & 3 \ 18 & -6 \end{array} \right]\]

Step 3: Multiply Matrix A by 2

Now, we perform scalar multiplication on matrix \(A\). Multiply each element of matrix \(A\) by 2:\[2A = 2 \times \left[ \begin{array}{rr} -4 & 4 \ 2 & -3 \ 1 & 5 \end{array} \right] = \left[ \begin{array}{rr} -8 & 8 \ 4 & -6 \ 2 & 10 \end{array} \right]\]

Step 4: Perform Matrix Subtraction (3B - 2A)

Subtract matrix \(2A\) from matrix \(3B\) by subtracting corresponding elements:\[3B - 2A = \left[ \begin{array}{rr} 21 & 0 \ 12 & 3 \ 18 & -6 \end{array} \right] - \left[ \begin{array}{rr} -8 & 8 \ 4 & -6 \ 2 & 10 \end{array} \right] = \left[ \begin{array}{rr} 21 - (-8) & 0 - 8 \ 12 - 4 & 3 - (-6) \ 18 - 2 & -6 - 10 \end{array} \right]\]Simplifying each element gives:\[= \left[ \begin{array}{rr} 29 & -8 \ 8 & 9 \ 16 & -16 \end{array} \right]\]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra, where every element of a matrix is multiplied by a single number, known as the scalar. This operation effectively scales the matrix without altering its shape or orientation. To perform scalar multiplication, you multiply every entry of the matrix by the scalar. For example, given the matrix \[B = \begin{bmatrix} 7 & 0 \ 4 & 1 \ 6 & -2 \end{bmatrix}\] and a scalar value of 3, the result of the scalar multiplication is \[3B = \begin{bmatrix} 21 & 0 \ 12 & 3 \ 18 & -6 \end{bmatrix}\].

• Maintain the same dimensions of the matrix after multiplication.
• Each element is independently multiplied.

This makes scalar multiplication straightforward and intuitive, allowing for quick scaling of matrices.

Matrix Addition

Matrix addition is the process by which two matrices of the same dimension are added together by summing their corresponding elements. For addition to be possible, the matrices involved must have the same number of rows and columns.
To illustrate, consider matrices \[A = \begin{bmatrix} -4 & 4 \ 2 & -3 \ 1 & 5 \end{bmatrix}\] and \[B = \begin{bmatrix} 7 & 0 \ 4 & 1 \ 6 & -2 \end{bmatrix}\]. Adding these matrices gives: \[A + B = \begin{bmatrix} -4 + 7 & 4 + 0 \ 2 + 4 & -3 + 1 \ 1 + 6 & 5 - 2 \end{bmatrix} = \begin{bmatrix} 3 & 4 \ 6 & -2 \ 7 & 3 \end{bmatrix}\].

• Ensure matrices are of the same dimension to perform addition.
• Sum each pair of corresponding elements from the matrices.

The simplicity and symmetry of matrix addition make it a useful tool in various mathematical models.

Matrix Subtraction

Matrix subtraction shares similarities with matrix addition and involves subtracting one matrix from another, element by element. This operation also requires that both matrices have the same dimensions. If we need to subtract matrix \(A\) from matrix \(B\), we do so by subtracting each corresponding element of \(A\) from \(B\).
For example, consider the matrices \[3B = \begin{bmatrix} 21 & 0 \ 12 & 3 \ 18 & -6 \end{bmatrix}\] and \[2A = \begin{bmatrix} -8 & 8 \ 4 & -6 \ 2 & 10 \end{bmatrix}\]. The operation \[3B - 2A\] involves: \[\begin{bmatrix} 21 - (-8) & 0 - 8 \ 12 - 4 & 3 - (-6) \ 18 - 2 & -6 - 10 \end{bmatrix} = \begin{bmatrix} 29 & -8 \ 8 & 9 \ 16 & -16 \end{bmatrix}\].

• Dimensions must match just like in matrix addition.
• Subtract correspondingly positioned elements from each matrix.

Matrix subtraction allows for the analysis of changes and differences between datasets.