More practice questions

5. Q5. Find a, b, c and x if the matrices below are equal.

   $$\begin{bmatrix} a+b & 3 \\ 2 & c \end{bmatrix} = \begin{bmatrix} 7 & x-1 \\ a-b & 2c-5 \end{bmatrix}$$
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   a + b = 7 and a − b = 2 give a = 9/2 and b = 5/2. 3 = x − 1 gives x = 4, and c = 2c − 5 gives c = 5.

6. Q6. For which values of p is the matrix below (i) diagonal; (ii) scalar; (iii) the identity matrix?

   $$\begin{bmatrix} p^2 & p-1 \\ 0 & p \end{bmatrix}$$
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   (i) Diagonal needs p − 1 = 0, so p = 1. (ii) Scalar also needs p² = p: with p = 1 this holds (1 = 1), so p = 1. (iii) Then both diagonal elements are 1, so p = 1 gives I. Only p = 1 works in every part.

7. Q7. How many 3 × 3 diagonal matrices have every element 0 or 1? How many of them are scalar?

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   The 6 elements off the diagonal must be 0; each of the 3 diagonal elements is 0 or 1, giving 2³ = 8 diagonal matrices. Scalar ones need equal diagonal elements: the zero matrix and I, so 2.

8. Q8. A square matrix of order 4 has how many elements on its diagonal and how many off it? Which positions (i, j) lie above the diagonal?

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   4 on the diagonal and 16 − 4 = 12 off it. Above the diagonal are the 6 positions with i < j: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4).