Question

If $\mathit{B}$is obtained be multiplying a column of$\mathit{A}$ by$\mathbf{9}$ , then the equation $\mathit{d}\mathit{e}\mathit{t}\mathit{B}\mathbf{=}\mathbf{9}\mathit{d}\mathit{e}\mathit{t}\mathit{A}$must hold.

Short answer

Therefore, the given condition is true.

Step-by-step solution

Definition

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

To check whether the given condition is true or false

By using the Laplace expansion along said column of,

We get:

$$\begin{gathered} A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \det A=1-0 \\ \det A=1 \end{gathered}$$

$$\begin{gathered} B=\left[\begin{array}{cc}9& 0\\ 0& 1\end{array}\right] \\ \det B=9-0 \\ \det B=9 \end{gathered}$$

role="math" localid="1664185768735" $$\begin{gathered} \det B=9\det A \\ \text{1 = 9(1)} \end{gathered}$$

Therefore, the given condition is true by using the Laplace expansion