Question

If $A = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  a&b&{ - 1}
\end{array}} \right]$ , then ${A^2} = $
A.Unit matrix
B.Null matrix
C.A
D.-A

Answer 1

Hint: We have given the matrix A now we have to find the nature of of the square of the matrix A, so simplifying the square of the matrix we will observe the nature of the square matrix, using the definition of matrices given i.e. unit matrix and null matrix to get the required answer.

Complete step-by-step answer:
Given data: $A = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  a&b&{ - 1}
\end{array}} \right]$
Solving for the matrix $A = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  a&b&{ - 1}
\end{array}} \right]$
Multiplying both sides by A
 $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  a&b&{ - 1}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  a&b&{ - 1}
\end{array}} \right]$
On multiplication and simplification
 $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  {a - a}&{b - b}&1
\end{array}} \right]$
On simplifying the terms of the matrix
 $ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right]$
Now we know that a $n \times n$ square matrix will be called an identity matrix of size n if all the non-diagonal elements of the matrix are zero and all the diagonal elements are one. This type of matrix is sometimes called a unit matrix of size n.
Therefore, we can say that ${A^2}$ is a unit matrix of size 3.
Hence, option (A) is correct

Note: In the question, we have mentioned some of the types of matrix let us discuss them
Unit matrix: A $n \times n$ square matrix will be called an identity matrix of size n if all the non-diagonal elements of the matrix are zero and all the diagonal elements are one. This type of matrix is sometimes called as a unit matrix of size n and is denoted by ${I_n}$ , where
 ${I_1} = \left[ 1 \right]$
 ${I_2} = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]$
 ${I_3} = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right]$
 ${I_4} = \left[ {\begin{array}{*{20}{c}}
  1&0&0&0 \\
  0&1&0&0 \\
  0&0&1&0 \\
  0&0&0&1
\end{array}} \right]$ …… and so on
Null matrix: It is a matrix having all of its entries as zero. It is also called a zero matrix, it can either be a rectangular or a square matrix. A null matrix is the identity element for the matrix addition.
 ${A_1} = \left[ 0 \right]$
 ${A_2} = \left[ {\begin{array}{*{20}{c}}
  0&0&0
\end{array}} \right]$
 \[{A_3} = \left[ {\begin{array}{*{20}{c}}
  0&0 \\
  0&0 \\
  0&0
\end{array}} \right]\]
 ${A_4} = \left[ {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  0 \\
  0
\end{array}} \right]$ , all are examples of null or zero matrices.