Skip to content
GATETECH
Official PYQ · 2000Q1.3Linear Algebra: Matrices, Determinants, System of Linear EquationsMCQ1 mark

GATE CS 2000 Q1.3 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Explain with AI

The determinant of the matrix [2000817220209061]\begin{bmatrix}2 &0 &0 &0 \\ 8& 1& 7& 2\\ 2& 0&2 &0 \\ 9&0 & 6 & 1 \end{bmatrix}

Choose one option
View worked solution

Answer: A

We are given the 4×44 \times 4 matrix:

A=[2000817220209061]A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 8 & 1 & 7 & 2 \\ 2 & 0 & 2 & 0 \\ 9 & 0 & 6 & 1 \end{bmatrix}

To find the determinant of AA, we can expand along the column or row containing the most zeros. Notice that the second column of AA has only one non-zero element, which is 11 at position (2,2)(2,2):

C2=[0100]C_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}

Expanding the determinant along the second column:

det⁡(A)=(−1)2+2⋅1⋅det⁡[200220961]\det(A) = (-1)^{2+2} \cdot 1 \cdot \det \begin{bmatrix} 2 & 0 & 0 \\ 2 & 2 & 0 \\ 9 & 6 & 1 \end{bmatrix}

The remaining 3×33 \times 3 matrix is a lower triangular matrix:

T=[200220961]T = \begin{bmatrix} 2 & 0 & 0 \\ 2 & 2 & 0 \\ 9 & 6 & 1 \end{bmatrix}

The determinant of any triangular matrix is simply the product of its diagonal elements:

det⁡(T)=2×2×1=4\det(T) = 2 \times 2 \times 1 = 4

Therefore:

det⁡(A)=1×4=4\det(A) = 1 \times 4 = 4

Thus, the correct option is A.

Free practice · Your answers on this page are not saved.

Published
Report a correction
Keep exploringMore Linear Algebra: Matrices, Determinants, System of Linear Equations questions