Question

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

Answer 1

Hint: A matrix with an equal number of rows and columns is called a square matrix. The square matrix of order $m$ is what mathematicians refer to as a $m \times m$ matrix. The order of the resulting matrix is unaffected by the addition or multiplication of any two square matrices i.e., $m \times m$.

Formula Used: Let a matrix $A = \left[ {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\
  {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\
  {{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\end{array}} \right]$ , then: ${A^2} = A \cdot A$ .

Complete step by step Solution:
We have matrix $A = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0&0 \\
  0&{ - 1}&0 \\
  0&0&{ - 1}
\end{array}} \right]$ .
To find the square of the given matrix $A$, we should apply the formula:
${A^2} = A \cdot A$
We can write this as follows:
${A^2} = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0&0 \\
  0&{ - 1}&0 \\
  0&0&{ - 1}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  { - 1}&0&0 \\
  0&{ - 1}&0 \\
  0&0&{ - 1}
\end{array}} \right]$
We get the squared matrix as:
${A^2} = \left[ {\begin{array}{*{20}{c}}
  {( - 1) \times ( - 1) + 0 \times 0 + 0 \times 0}&{( - 1) \times 0 + 0 \times ( - 1) + 0 \times 0}&{( - 1) \times 0 + 0 \times 0 + 0 \times ( - 1)} \\
  {0 \times ( - 1) + ( - 1) \times 0 + 0 \times 0}&{0 \times 0 + ( - 1) \times ( - 1) + 0 \times 0}&{0 \times 0 + ( - 1) \times 0 + 0 \times ( - 1)} \\
  {0 \times ( - 1) + 0 \times 0 + ( - 1) \times 0}&{0 \times 0 + 0 \times ( - 1) + ( - 1) \times 0}&{0 \times 0 + 0 \times 0 + ( - 1) \times ( - 1)}
\end{array}} \right]$
Hence, the answer will be ${A^2} = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right] = I$ , where $I$ is a unit matrix.

Therefore, the correct option is (B).

Note: To multiply two matrices, we must first determine whether the first matrix's number of columns and rows equals the second matrix's number of rows. Then add all of the results after multiplying each element of the first matrix's column by each element of the second matrix's row.