GATE CS 2015 Set 2 Q.37 (CS Q.27): Linear Algebra

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

(i) Add the third row to the second row.
(ii) Subtract the third column from the first column.

The determinant of the resultant matrix is _____.

Your answer

Official answer and solution

Solution

Observe linear dependence among columns before performing elementary transformations.

Consider the given matrix:

A = \begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}

Examine the first and third columns:

C_1 = \begin{bmatrix} 3 \\ 7 \\ 13 \end{bmatrix},\quad C_3 = \begin{bmatrix} 45 \\ 105 \\ 195 \end{bmatrix} = 15 \begin{bmatrix} 3 \\ 7 \\ 13 \end{bmatrix} = 15 C_1.

Because $C_3$ is an exact scalar multiple of $C_1$ , the columns of $A$ are linearly dependent, so $\det(A) = 0$ .

The operations performed on the matrix are:
(i) $R_2 \leftarrow R_2 + R_3$
(ii) $C_1 \leftarrow C_1 - C_3$

Elementary operations of the form $R_i \leftarrow R_i + c R_j$ and $C_i \leftarrow C_i + c C_j$ do not change the determinant of a matrix:

\det(\text{resultant}) = \det(A) = 0.

Final answer

Common trap

Manually computing the full determinant expansion with large intermediate numbers rather than recognizing $C_3 = 15 C_1$ immediately.

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