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Official PYQ · 2015Set 2 · Q27Linear Algebra: Matrices, Determinants, System of Linear EquationsNAT1 mark

GATE CS 2015 Set 2 Q27 — Linear Algebra: Matrices, Determinants, System of Linear Equations

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Perform the following operations on the matrix [344579105132195]\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 _____.

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Answer: 0

Elementary row operations of the form Ri←Ri+kRjR_i \leftarrow R_i + k R_j and column operations of the form Ci←Ci+kCjC_i \leftarrow C_i + k C_j do not alter the determinant of a matrix.

Therefore, the determinant of the resultant matrix is exactly equal to the determinant of the original matrix AA:

A=[344579105132195]A = \begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}

Let us examine the relationship between the first column (C1C_1) and the third column (C3C_3):

C1=[3713]andC3=[45105195]=15[3713]=15C1C_1 = \begin{bmatrix} 3 \\ 7 \\ 13 \end{bmatrix} \quad \text{and} \quad C_3 = \begin{bmatrix} 45 \\ 105 \\ 195 \end{bmatrix} = 15 \begin{bmatrix} 3 \\ 7 \\ 13 \end{bmatrix} = 15 C_1

Since the third column is a scalar multiple of the first column (C3=15C1C_3 = 15 C_1), the columns of the matrix are linearly dependent. Consequently, the determinant of the matrix is 00.

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