Question

If $A = \begin{bmatrix} {1}&{-3}\\ {2}&{K} \\ \end{bmatrix}$ and $A^2-14A+10I=A$,then $K$ = is equal to :

• A. 1 or 4
• B. 4 and not 1
• C. -4 .
• D. 0

Answer 1

The Correct Option is B

\(A = \begin{bmatrix}1&-3\\ 2&k\end{bmatrix}\)
\(\therefore A^{2} -4A +10I = A\)
\(\Rightarrow \begin{bmatrix}1&-3\\ 2&k\end{bmatrix} \begin{bmatrix}1&-3\\ 2&k\end{bmatrix}-4 \begin{bmatrix}1&-3\\ 2&k\end{bmatrix}\)
\(+ 10 \begin{bmatrix}1&0\\ 0&1\end{bmatrix} = \begin{bmatrix}1&-3\\ 2&k\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}-5&-3-3k\\ 2+2k& -6+k^{2}\end{bmatrix} - \begin{bmatrix}4&-12\\ 8&4k\end{bmatrix} + \begin{bmatrix}10&0\\ 0&10\end{bmatrix} = \begin{bmatrix}1&-3\\ 2&k\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}1&9-3k\\ -6+2k&4+k^{2}-4k\end{bmatrix} =\begin{bmatrix}1&-3\\ 2&k\end{bmatrix}\)
\(\Rightarrow 9 -3 k = -3, - 6+2k = 2 \,\,\,\,\dots(i)\)
and \(4 + k^{2} - 4k = k\)
\(\Rightarrow \, k^{2}- 5k + 4 = 0\)
\(\Rightarrow\, \left(k - 4\right) \left(k - 1\right) = 0\)
\(\Rightarrow \, k = 4,1\) 
The term "matrix" refers to a rectangular arrangement of m and n elements where the arrangement is made up of m rows and n columns contained in square brackets.

Few types of matrices are–

Column Matrix 

A column matrix is a matrix with just one column. In general, we may state that the number of rows in the column matrix is 0, but the number of columns is 1. 

Row Matrix 

A row matrix is a matrix with just one row. In the row matrix, there are typically 1 row and 0 columns.

Square Matrix 

A square matrix is one that has the same number of rows and columns as rows. If a matrix is m*n in size, then m=n is always present in the square matrix.

Diagonal Matrix 

A matrix that solely contains elements in diagonal positions is referred to as a diagonal matrix.

Zero Matrix 

A matrix having a Zero on all the positions then It is called a Zero Matrix.

Scalar Matrix 

A diagonal Matrix having the same elements on diagonal Position Then It is called as a Scalar Matrix.

Identity Matrix 

In the Square Matrix, Elements which are halted on diagonal positions are 1 and rest elements are 0 called an Identity Matrix.