Question

Give example, with justification , of the following: (1) two non -zero ,3×3 matrices A and B , with|A| =0, |B|= (5/7)i ; (2) . two non - singular 2×2 matrices C and D , with |C| = √2 |D| ?

Answer 1

Answer on Question #74598 – Math – Linear Algebra

Question

Give example, with justification, of the following:

(1) two non-zero, $3 \times 3$ matrices $A$ and $B$, with $|A| = 0$, $|B| = \frac{5}{7} i$;

Solution

Let’s consider the following non-zero matrices (all elements of zero-matrix are zeroes)

$$A = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 1 & 2 & 3 \end{array} \right), \quad B = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & i & 2 \\ 0 & \frac{i}{7} & 1 \end{array} \right).$$ $$|A| = \det(A) = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 1 & 2 & 3 \end{array} \right| = 0 \quad \text{because the matrix } A \text{ has linearly dependent rows: } (2; 2; 2) = 2 \cdot (1; 1; 1).$$ $$|B| = \left| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & i & 2 \\ 0 & \frac{i}{7} & 1 \end{array} \right| = 1 \cdot \left| \begin{array}{cc} i & 2 \\ \frac{i}{7} & 1 \end{array} \right| = i - \frac{2i}{7} = i \left(1 - \frac{2}{7}\right) = \frac{5}{7} i.$$

Question

Give example, with justification, of the following:

(2) two non-singular $2 \times 2$ matrices $C$ and $D$, with $|C| = \sqrt{2} \cdot |D|$.

Solution

Let’s consider the following non-singular matrices (the determinant of a singular matrix is equal to zero)

$$D = \left( \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right), \quad C = \left( \begin{array}{cc} 2\sqrt{2} & 1 \\ \sqrt{2} & 1 \end{array} \right).$$ $$|D| = \left| \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right| = 2 - 1 = 1; \quad |C| = \left| \begin{array}{cc} 2\sqrt{2} & 1 \\ \sqrt{2} & 1 \end{array} \right| = 2\sqrt{2} - \sqrt{2} = \sqrt{2};$$

So $|C| = \sqrt{2} \cdot |D|$.

Answer:

Example (1): $A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix}$, $B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & i & 2 \\ 0 & \frac{i}{7} & 1 \end{pmatrix}$.

Example (2): $D = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, $C = \begin{pmatrix} 2\sqrt{2} & 1 \\ \sqrt{2} & 1 \end{pmatrix}$.

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