Question

There exists a real number 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 is invertible

So, the given statement is true.

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

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 a “ 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

$detA=-6.\left|\begin{array}{ccc}1& 2& 4\\ 5& 6& 7\\ 8& 9& 7\end{array}\right|+5.\left|\begin{array}{ccc}1& 2& 3\\ 5& 6& k\\ 8& 9& 8\end{array}\right|$

$$\begin{gathered} A=-6·9+5·(7k-41) \\ A=35k-259 \end{gathered}$$

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