Question

The value of λ for which the matrix \(\begin{pmatrix} 1 & 0 & λ \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}\) is a singular matrix is :

• A. 0
• B. -1
• C. 1
• D. 2

Answer 1

The Correct Option is C

The given matrix is:\[\begin{pmatrix} 1 & 0 & λ \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}\]To determine the value of λ for which the matrix is singular, we must find when its determinant is zero. A matrix is singular if and only if its determinant is zero.
For a 3x3 matrix \[\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\], the determinant is given by:
\[a(ei - fh) - b(di - fg) + c(dh - eg)\]
Applying this to our matrix, we calculate the determinant:
\[\begin{vmatrix} 1 & 0 & λ \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{vmatrix} = 1(1 \cdot 1 - 0 \cdot 0) - 0(0 \cdot 1 - 0 \cdot 1) + λ(0 \cdot 0 - 1 \cdot 1)\]
This simplifies to:
\[1(1) - 0 + λ(-1) = 1 - λ\]
For the matrix to be singular, the determinant must equal zero:
\[1 - λ = 0\]
Solving for λ, we have:
\[λ = 1\]
Therefore, the value of λ that makes the matrix singular is 1.