Matrices

Practice sheet

Fresh questions from the whole chapter, to do on paper.

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  1. Let A = (3−375)\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix} and B = (−4−4−55)\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?
  2. Let A = (2062)\begin{pmatrix} 2 & 0 \\ 6 & 2 \end{pmatrix}. What is the entry in row 2, column 1 of 3A?
  3. A shop records sales in a matrix with one row for each of its 4 stores and one column for each of 7 days. What are the dimensions of the matrix?
    1. 4 × 7
    2. 7 × 4
    3. 28 × 1
  4. Let A = (423−2)\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix} and B = (7954)\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}. What is the entry in row 1, column 1 of AB?
  5. Let A = (8884)\begin{pmatrix} 8 & 8 \\ 8 & 4 \end{pmatrix} and B = (2332)\begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}. What is the entry in row 2, column 1 of AB?
  6. A is a 2 × 5 matrix and B is a 5 × 8 matrix. What are the dimensions of AB?
    1. 2 × 8
    2. 5 × 5
    3. AB is not defined
  7. What is the determinant of (−2−584)\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}?
  8. Let A = (5318)\begin{pmatrix} 5 & 3 \\ 1 & 8 \end{pmatrix}. What is the entry in row 1, column 1 of the inverse of A?
  9. For which value of k does (k3102)\begin{pmatrix} k & 3 \\ 10 & 2 \end{pmatrix} have no inverse?
  10. Let A = (9236)\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}. What is the first component of A(−3−1)\begin{pmatrix} -3 \\ -1 \end{pmatrix}?
  11. The matrix (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} maps the point (8, 0) to (0, 8) and (0, 8) to (−8, 0). Which transformation does it represent?
    1. A rotation of 90° counterclockwise about the origin
    2. A reflection in the line y = x
    3. A rotation of 90° clockwise about the origin
  12. A shape with an area of 2 square units is transformed by the matrix (2223)\begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}. What is the area of the image?