Math Problem Statement

Find the following matrices where A = [[-2, 5], [2, 7], [2, -9]] and B = [[-2, -6], [7, 9], [-1, -4]], and solve for 8A - 6B.

Solution

To solve for 8A6B8A - 6B, first find 8A8A and 6B6B, then perform the subtraction.

Given: A=[252729]A = \begin{bmatrix} -2 & 5 \\ 2 & 7 \\ 2 & -9 \end{bmatrix} B=[267914]B = \begin{bmatrix} -2 & -6 \\ 7 & 9 \\ -1 & -4 \end{bmatrix}

Calculate 8A8A: 8A=8×[252729]=[8×28×58×28×78×28×9]=[164016561672]8A = 8 \times \begin{bmatrix} -2 & 5 \\ 2 & 7 \\ 2 & -9 \end{bmatrix} = \begin{bmatrix} 8 \times -2 & 8 \times 5 \\ 8 \times 2 & 8 \times 7 \\ 8 \times 2 & 8 \times -9 \end{bmatrix} = \begin{bmatrix} -16 & 40 \\ 16 & 56 \\ 16 & -72 \end{bmatrix}

Calculate 6B6B: 6B=6×[267914]=[6×26×66×76×96×16×4]=[12364254624]6B = 6 \times \begin{bmatrix} -2 & -6 \\ 7 & 9 \\ -1 & -4 \end{bmatrix} = \begin{bmatrix} 6 \times -2 & 6 \times -6 \\ 6 \times 7 & 6 \times 9 \\ 6 \times -1 & 6 \times -4 \end{bmatrix} = \begin{bmatrix} -12 & -36 \\ 42 & 54 \\ -6 & -24 \end{bmatrix}

Subtract 6B6B from 8A8A: 8A6B=[164016561672][12364254624]=[16+1240+361642565416+672+24]=[4762622248]8A - 6B = \begin{bmatrix} -16 & 40 \\ 16 & 56 \\ 16 & -72 \end{bmatrix} - \begin{bmatrix} -12 & -36 \\ 42 & 54 \\ -6 & -24 \end{bmatrix} = \begin{bmatrix} -16 + 12 & 40 + 36 \\ 16 - 42 & 56 - 54 \\ 16 + 6 & -72 + 24 \end{bmatrix} = \begin{bmatrix} -4 & 76 \\ -26 & 2 \\ 22 & -48 \end{bmatrix}

So, 8A6B=[4762622248]8A - 6B = \begin{bmatrix} -4 & 76 \\ -26 & 2 \\ 22 & -48 \end{bmatrix}.

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Math Problem Analysis

Mathematical Concepts

Matrix multiplication
Matrix subtraction

Formulas

Matrix multiplication: kA = k * A (where k is a scalar and A is a matrix)
Matrix subtraction: A - B = [[a11 - b11, a12 - b12], ...]

Theorems

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Suitable Grade Level

Grades 10-12