Question

If \[A = dig(2, - 1,3)\], $B = dig( - 1,3,2)$ then ${A^2}B = $
A $dig(5,4,11)$
B $dig( - 4,3,18)$
C $dig(3,1,8)$
D B

Answer 1

Hint: First we will convert given diagonal matrix A and B into $3 \times 3$ matrix. Then will find the ${A^2}$ using the product of matrices. Then multiply the resultant matrix ${A^2}$ with matrix $B$. Then convert the $3 \times 3$ matrix into a diagonal matrix.

Complete step by step Solution:
We know that $dig(a,b,c) = \left[ {\begin{array}{*{20}{c}}
  a&0&0 \\
  0&b&0 \\
  0&0&c
\end{array}} \right]$
Converting given A diagonal matrix into $3 \times 3$ matrix
$A = \left[ {\begin{array}{*{20}{c}}
  2&0&0 \\
  0&{ - 1}&0 \\
  0&0&3
\end{array}} \right]$
$B = dig( - 1,3,2)$
Converting given diagonal matrix B into $3 \times 3$ matrix
$B = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0&0 \\
  0&3&0 \\
  0&0&2
\end{array}} \right]$
We know that ${A^2} = A.A$
${A^2} = \left[ {\begin{array}{*{20}{c}}
  2&0&0 \\
  0&{ - 1}&0 \\
  0&0&3
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  2&0&0 \\
  0&{ - 1}&0 \\
  0&0&3
\end{array}} \right]$
After multiplication, we will get
${A^2} = \left[ {\begin{array}{*{20}{c}}
  {4 + 0 + 0}&0&0 \\
  0&{0 + 1 + 0}&0 \\
  0&0&{0 + 0 + 9}
\end{array}} \right]$
After solving, we get
${A^2} = \left[ {\begin{array}{*{20}{c}}
  4&0&0 \\
  0&1&0 \\
  0&0&9
\end{array}} \right]$
\[{A^2}B = \left[ {\begin{array}{*{20}{c}}
  4&0&0 \\
  0&1&0 \\
  0&0&9
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  { - 1}&0&0 \\
  0&3&0 \\
  0&0&2
\end{array}} \right]\]
After multiplication, we will get
${A^2}B = \left[ {\begin{array}{*{20}{c}}
  { - 4 + 0 + 0}&0&0 \\
  0&{0 + 3 + 0}&0 \\
  0&0&{0 + 0 + 18}
\end{array}} \right]$
After solving this, we get
${A^2}B = \left[ {\begin{array}{*{20}{c}}
  { - 4}&0&0 \\
  0&3&0 \\
  0&0&{18}
\end{array}} \right]$
Converting $3 \times 3$ matrix into diagonal matrix
${A^2}B = dig( - 4,3,18)$

Therefore, the correct option is (B).

Note:Students should know how to convert the diagonal matrix into $3 \times 3$ matrix correctly to avoid any mistakes. And do calculations correctly to get the correct required solution.