Matrices
Write your working on paper. Reveal each answer only after you've had a go.
1. Given \(A = \begin{bmatrix} 8 & 5 & 3 \\ 0 & 9 & 9 \\ 7 & -1 & 6 \end{bmatrix}\) and \(B = \begin{bmatrix} 8 & 3 & 5 \\ 5 & -3 & -4 \\ 7 & -4 & -4 \end{bmatrix}\), calculate each of the following.
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\(A + B\)
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Add corresponding elements.\(\begin{bmatrix} 16 & 8 & 8 \\ 5 & 6 & 5 \\ 14 & -5 & 2 \end{bmatrix}\) -
\(B - A\)
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Subtract corresponding elements.\(\begin{bmatrix} 0 & -2 & 2 \\ 5 & -12 & -13 \\ 0 & -3 & -10 \end{bmatrix}\) -
\(A \times B\)
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Row-by-column: each entry is a row of A dotted with a column of B.\(\begin{bmatrix} 110 & -3 & 8 \\ 108 & -63 & -72 \\ 93 & 0 & 15 \end{bmatrix}\) -
\(B \times A\)
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Note AB and BA are generally different.\(\begin{bmatrix} 99 & 62 & 81 \\ 12 & 2 & -36 \\ 28 & 3 & -39 \end{bmatrix}\) -
\(|A|\)
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Expand along the first row using 2x2 minors.\(|A| = 630\) -
\(|B|\)
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Expand along the first row using 2x2 minors.\(|B| = -51\)
2. Solve each set of simultaneous equations using matrices.
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Solve the simultaneous equations:
\(-4x - 5y - 5z = 13\)
\(-4x - 4y - 5z = 12\)
\(4x - 2y - z = 18\)Show answer
Write them in matrix form \(\begin{bmatrix} -4 & -5 & -5 \\ -4 & -4 & -5 \\ 4 & -2 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 13 \\ 12 \\ 18 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = 3, y = -1, z = -4 -
Solve the simultaneous equations:
\(3x + y - 5z = 6\)
\(-x - 4y + 2z = 11\)
\(4x - 3y + 3z = -23\)Show answer
Write them in matrix form \(\begin{bmatrix} 3 & 1 & -5 \\ -1 & -4 & 2 \\ 4 & -3 & 3 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 11 \\ -23 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = -5, y = -4, z = -5