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Mathematics Question on Algebra of Complex Numbers

If A=[1−3 2K ]A = \begin{bmatrix} {1}&{-3}\\\ {2}&{K} \\\ \end{bmatrix} and A2−14A+10I=AA^2-14A+10I=A,then KK = is equal to :

A

1 or 4

B

4 and not 1

C

-4 .

D

0

Answer

4 and not 1

Explanation

Solution

A=[1−3 2k]A = \begin{bmatrix}1&-3\\\ 2&k\end{bmatrix}
∴A2−4A+10I=A\therefore A^{2} -4A +10I = A
⇒[1−3 2k][1−3 2k]−4[1−3 2k]\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[10 01]=[1−3 2k]+ 10 \begin{bmatrix}1&0\\\ 0&1\end{bmatrix} = \begin{bmatrix}1&-3\\\ 2&k\end{bmatrix}
⇒[−5−3−3k 2+2k−6+k2]−[4−12 84k]+[100 010]=[1−3 2k]\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}
⇒[19−3k −6+2k4+k2−4k]=[1−3 2k]\Rightarrow \begin{bmatrix}1&9-3k\\\ -6+2k&4+k^{2}-4k\end{bmatrix} =\begin{bmatrix}1&-3\\\ 2&k\end{bmatrix}
⇒9−3k=−3,−6+2k=2    …(i)\Rightarrow 9 -3 k = -3, - 6+2k = 2 \,\,\,\,\dots(i)
and 4+k2−4k=k4 + k^{2} - 4k = k
⇒ k2−5k+4=0\Rightarrow \, k^{2}- 5k + 4 = 0
⇒ (k−4)(k−1)=0\Rightarrow\, \left(k - 4\right) \left(k - 1\right) = 0
⇒ k=4,1\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.