Question

If A= 3 1, -1 2 find the value of A^2 -5A + 7I where
I is unit matrix of order 2
​

Answer 1

$\large\underline{\sf{Solution-}}$

• Given

$\rm :\longmapsto\:A = \begin{bmatrix} 3 & 1\\ - 1 & 2\end{bmatrix}$

Now,

• Consider,

$\rm :\longmapsto\: {A}^{2} = A \times A$

$\rm :\longmapsto\: = \: \begin{bmatrix} 3 & 1\\ - 1 & 2\end{bmatrix}\begin{bmatrix} 3 & 1\\ - 1 & 2\end{bmatrix}$

$\rm :\longmapsto\: = \: \begin{bmatrix} 9 - 1 & 3 + 2\\ - 3 - 2 & - 1 + 4\end{bmatrix}$

$\rm :\longmapsto\: = \: \begin{bmatrix} 8 & 5\\ - 5 & 3\end{bmatrix}$

$\rm :\implies\: \boxed{ \bf \: {A}^{2} \: = \: \begin{bmatrix} 8 & 5\\ - 5 & 3\end{bmatrix}}$

Now,

• Consider,

$\rm :\longmapsto\: {A}^{2} - 5A + 7I$

$\rm :\longmapsto\: = \: \begin{bmatrix} 8 & 5\\ - 5 & 3\end{bmatrix} - 5\begin{bmatrix} 3 & 1\\ - 1 & 2\end{bmatrix} + 7\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$

$\rm :\longmapsto\: = \: \begin{bmatrix} 8 & 5\\ - 5 & 3\end{bmatrix} - \begin{bmatrix} 15 & 5\\ - 5 & 10\end{bmatrix} + \begin{bmatrix} 7 & 0\\ 0 & 7\end{bmatrix}$

$\rm :\longmapsto\: = \: \begin{bmatrix} 8 - 15 + 7 & 5 - 5\\ - 5 + 5 & 3 - 10 + 7\end{bmatrix}$

$\rm :\longmapsto\: = \: \begin{bmatrix} 0 & 0\\ 0 & 0\end{bmatrix}$

$\rm :\implies\: \boxed{ \bf{ \: {A}^{2} - 5A + 7I = 0}}$

Additional Information :-

Row Matrix : A matrix having only one row is called a row matrix. 

Column Matrix : A matrix having only one column is called a column matrix. 

Zero or Null Matrix : If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by 0. 

Singleton Matrix : If in a matrix there is only element then it is called singleton matrix.

Square Matrix : If the number of rows and the number of columns in a matrix are equal, then it is called a square matrix.

Diagonal Matrix : If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a diagonal matrix.

Scalar Matrix : If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix.

Unit Matrix or Identity Matrix : If all the elements of a principal diagonal in a diagonal matrix are 1, then it is called a unit matrix. A unit matrix of order n is denoted by I.