Matrices
Write your working on paper. Reveal each answer only after you've had a go.
1. Given \(A = \begin{bmatrix} 8 & -4 & 9 \\ 4 & -4 & 3 \\ -5 & 0 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 4 & 4 & 3 \\ 0 & -5 & 8 \\ -5 & 5 & 9 \end{bmatrix}\), calculate each of the following.
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\(A + B\)
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Add corresponding elements.\(\begin{bmatrix} 12 & 0 & 12 \\ 4 & -9 & 11 \\ -10 & 5 & 9 \end{bmatrix}\) -
\(B - A\)
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Subtract corresponding elements.\(\begin{bmatrix} -4 & 8 & -6 \\ -4 & -1 & 5 \\ 0 & 5 & 9 \end{bmatrix}\) -
\(A \times B\)
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Row-by-column: each entry is a row of A dotted with a column of B.\(\begin{bmatrix} -13 & 97 & 73 \\ 1 & 51 & 7 \\ -20 & -20 & -15 \end{bmatrix}\) -
\(B \times A\)
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Note AB and BA are generally different.\(\begin{bmatrix} 33 & -32 & 48 \\ -60 & 20 & -15 \\ -65 & 0 & -30 \end{bmatrix}\) -
\(|A|\)
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Expand along the first row using 2x2 minors.\(|A| = -120\) -
\(|B|\)
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Expand along the first row using 2x2 minors.\(|B| = -575\)
2. Solve each set of simultaneous equations using matrices.
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Solve the simultaneous equations:
\(-3x - 3y + 2z = -2\)
\(-3x - 2y + 4z = 1\)
\(-2x - y - 5z = 10\)Show answer
Write them in matrix form \(\begin{bmatrix} -3 & -3 & 2 \\ -3 & -2 & 4 \\ -2 & -1 & -5 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -2 \\ 1 \\ 10 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = -5, y = 5, z = -1 -
Solve the simultaneous equations:
\(3x + 5y - 2z = 41\)
\(-5x + y - z = -18\)
\(4x + 3y + 4z = 20\)Show answer
Write them in matrix form \(\begin{bmatrix} 3 & 5 & -2 \\ -5 & 1 & -1 \\ 4 & 3 & 4 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 41 \\ -18 \\ 20 \end{bmatrix}\), then solve (invert the coefficient matrix, or use a calculator).x = 5, y = 4, z = -3