Question

There exists a real numberK such that the matrix$\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& k& 7\\ 8& 9& 8& 7\\ 0& 0& 6& 5\end{array}\right]$ is invertible.

Short answer

Therefore,

$k=3252$and it isinvertible

So, the given statement is true.

Step-by-step solution

Matrix Definition

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “$m\times n$ .”

Given

Given matrix,

$A=\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& k& 7\\ 8& 9& 8& 7\\ 0& 0& 6& 5\end{array}\right]$

To check whether the given condition is true or false

We compute

$detA=-6\cdot \left|\begin{array}{ccc}1& 2& 3\\ 5& 6& 7\\ 8& 9& 7\end{array}\right|+5\left|\begin{array}{ccc}1& 2& 3\\ 5& 6& k\\ 8& 9& 8\end{array}\right|$

role="math" localid="1659527820355" $$\begin{gathered} A=-6\cdot 9+5\cdot (7k-41) \\ A=35k-259. \end{gathered}$$

So, for example,A is invertible for $k=3252$.