Question

Among the following, the matrices with non-zero determinant are 

• A. P, Q and R
• B. P, R and S
• C. P, Q and S
• D. Q, R and S

Hint:

For diagonal or triangular matrices, the determinant equals the product of diagonal elements. If any row (or column) is linearly dependent on others, the determinant becomes zero.

Answer 1

The Correct Option is A

Step 1: Determinant of P.
$P$ is an identity matrix. \[ \det(P) = 1 \ (\text{non-zero}) \] Step 2: Determinant of Q.
$Q$ is a diagonal matrix. \[ \det(Q) = 1 \times 2 \times 3 \times 4 = 24 \ (\text{non-zero}) \] Step 3: Determinant of R.
$R$ is a lower triangular matrix with diagonal elements 1, 2, 3, 4. \[ \det(R) = 1 \times 2 \times 3 \times 4 = 24 \ (\text{non-zero}) \] Step 4: Determinant of S.
For $S$, rows are linearly dependent (each can be written as a linear combination of others). Hence, \[ \det(S) = 0. \] Step 5: Conclusion.
Matrices with non-zero determinants are P, Q, and R. Thus, the correct answer is (A).