Question-4

Let \(A=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\ ta_{11}-sa_{31} & ta_{12}-sa_{32} & ta_{13}-sa_{33}\\ ra_{31} & ra_{32} & ra_{33} \end{bmatrix}\) be a matrix and \(r,s,t\neq0\). Find \(\text{det}(A)\).

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Consider the matrix \(B\):

\[ B=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\ 0 & 0 & 0\\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

We can now do the following row operations on \(B\):

\[ \begin{aligned} R_{2} & \rightarrow R_{2}+tR_{1}-sR_{3}\\ R_{3} & \rightarrow rR_{3} \end{aligned} \]

These two row operations will give us \(A\). The determinants are related as follows:

\[ \text{det}(A)=r\cdot\text{det}(B) \]

Since \(B\) has a zero row, its determinant is zero. Hence \(\text{det}(A)=0\).