CMI-2018-DataScience-A: 2

Question

Let $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}, C=\begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$ and $D=\begin{bmatrix} -1 & 2 & 3 \\ 4 & -5 & 6 \\ 7 & 8 & -9 \end{bmatrix}$.

Which of the following statements are true?

• A. $|A|=|B|$
• B. $|C|=|D|$
• C. $|B|=-|C|$
• D. $|A|=-|D|$

Answer 1

|A| = |B| = |C| = 0 != |D|

 

Ans. A

Comments

What about option C?

0 = -0

Answer 2

As we know the property of determinant, $|A^T|=|A|$.

$|B|$ is the transpose of $|A|$

$\therefore |B|=|A^T|$

in the same way, if two rows(or columns ) of a determinant are interchanged, the sign of the value of the determinant is also changed.

Form $[A]$ if we interchange $R_1.R_2$ we get $[C]$

$\therefore |A|=|B|=|C|=0,|D|\neq0$

Answer 3

Matrix B is the transpose of matrix A.

Therefore, by determinant properties, |B| = |A| (as |A| = |A’|).

Interchanging rows R1 and R2 in |C| we get -|A| (again det properties, when we interchange rows or columns, det changes sign but value is same).

Therefore,  |C| = -|A| => |C| = -|B|

Hence both options a and c are correct.