Matrices
Write your working on paper. Reveal each answer only after you've had a go.
1. Given \(A = \begin{bmatrix} 9 & 7 & 8 \\ 1 & -3 & 8 \\ 0 & 7 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 7 & 2 & 7 \\ 2 & 5 & -2 \\ 1 & 5 & 4 \end{bmatrix}\), calculate each of the following.
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\(A + B\)
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Add corresponding elements.\(\begin{bmatrix} 16 & 9 & 15 \\ 3 & 2 & 6 \\ 1 & 12 & 4 \end{bmatrix}\) -
\(B - A\)
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Subtract corresponding elements.\(\begin{bmatrix} -2 & -5 & -1 \\ 1 & 8 & -10 \\ 1 & -2 & 4 \end{bmatrix}\) -
\(A \times B\)
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Row-by-column: each entry is a row of A dotted with a column of B.\(\begin{bmatrix} 85 & 93 & 81 \\ 9 & 27 & 45 \\ 14 & 35 & -14 \end{bmatrix}\) -
\(B \times A\)
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Note AB and BA are generally different.\(\begin{bmatrix} 65 & 92 & 72 \\ 23 & -15 & 56 \\ 14 & 20 & 48 \end{bmatrix}\) -
\(|A|\)
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Expand along the first row using 2x2 minors.\(|A| = -448\) -
\(|B|\)
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Expand along the first row using 2x2 minors.\(|B| = 225\)
2. Solve each set of simultaneous equations using matrices.
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Solve the simultaneous equations:
\(4x - 4y + 2z = 18\)
\(2x + 5y + 2z = 35\)
\(5x + 2y - 3z = 16\)Show answer
Write them in matrix form \(\begin{bmatrix} 4 & -4 & 2 \\ 2 & 5 & 2 \\ 5 & 2 & -3 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 18 \\ 35 \\ 16 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = 5, y = 3, z = 5 -
Solve the simultaneous equations:
\(-2x + y + 4z = 10\)
\(x + 4y - 5z = -26\)
\(3x + 4y - 2z = -33\)Show answer
Write them in matrix form \(\begin{bmatrix} -2 & 1 & 4 \\ 1 & 4 & -5 \\ 3 & 4 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 10 \\ -26 \\ -33 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = -5, y = -4, z = 1