Question

B = [ 0 a 0 0]
b 0 0 0
0 0 c 0
0 0 0 d
Let Bn the ( n x n) submatrix in the TOP left hand corner of B. Define B1, B2, B3 and B4. Compute determinate of B1, B2 , B3 and B4. Find conditions of a, b, c, d such that 4 determinants cannot be negative .

Answer 1

Answer on Question #63634 – Math – Linear Algebra

Question

B = [0 a 0 0]

b 0 0 0

0 0 c 0

0 0 0 d

Let Bn the (n x n) submatrix in the TOP left hand corner of B. Define B1, B2, B3 and B4. Compute determinate of B1, B2, B3 and B4. Find conditions of a, b, c, d such that 4 determinants cannot be negative.

Solution

$B1 = (0)$ is $1 \times 1$ matrix, $\det(B1) = 0$ for all values of $a, b, c, d$.

$B2 = \begin{pmatrix} 0 & a \\ b & 0 \end{pmatrix}$ is $2 \times 2$ matrix, $\det(B2) = 0 \cdot 0 - b \cdot a = -ab$. So $\det(B2) \geq 0$ is equivalent to $ab \leq 0$.

$B3 = \begin{pmatrix} 0 & a & 0 \\ b & 0 & 0 \\ 0 & 0 & c \end{pmatrix}$ is $3 \times 3$ matrix, $\det(B3) = c \cdot \det(B2) = -abc$. So $\det(B3) \geq 0$ is equivalent to $abc \leq 0$.

$B4 = \begin{pmatrix} 0 & a & 0 & 0 \\ b & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{pmatrix}$ is $4 \times 4$ matrix, $\det(B4) = d \cdot \det(B3) = -abcd$. So $\det(B4) \geq 0$ is equivalent to $abcd \leq 0$.

These 4 determinants cannot be negative if one of the following conditions holds:

1) $ab = 0$;

2) $(ab < 0)$ and $(c = 0)$;

3) $(ab < 0)$ and $(c > 0)$ and $(d \geq 0)$.

**Answer:** 1) $ab = 0$; 2) $(ab < 0)$ and $(c = 0)$; 3) $(ab < 0)$ and $(c > 0)$ and $(d \geq 0)$.

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