2k views

Perform the following operations on the matrix $\begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}$

1. Add the third row to the second row
2. Subtract the third column from the first column.

The determinant of the resultant matrix is _____.

| 2k views

Answer : $-$$0$, because it is easy to see that first column and third column are multiple of each other.

Third column = First column $^{\ast }$ $15.$

So rank is  $< 3$, so Determinant must be $0$.

It stays zero as row & column transformations don't affect determinant.
by Boss (41.6k points)
edited
0
its *15
+2
But note that, interchanging the rows and columns effects the sign !

A =$\begin{bmatrix} 7 & 4\\ 9 & 3 \end{bmatrix}$ ==> |A| = 21-36=-15

interchange the rows

B =$\begin{bmatrix} 9 & 3 \\ 7 & 4 \end{bmatrix}$ ==> |A| = 36-21=15
Ans Zero, row and column transformations doesn't affect determinant.
by Boss (13.5k points)
0
pls anyone show the matrix after transformation
+1 vote
ZERO
by (21 points)
Only interchanging two rows or two columns determinant value becomes -ve .Except any elementary operation doesn't affect determinant.

by Active (4.7k points)