\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {{\rm{x}} + {\rm{y}}}&{\rm{y}}\\ {\rm{x}}&{{\rm{x}} - {\rm{y}}} \end{array}} \right],{\rm{\;B}} = \left[ {\begin{array}{*{20}{c}} 3\\ { - 2} \end{array}} \right]\) and \({\rm{C}} = \left[ {\begin{array}{*{20}{c}} 4\\ { - 2} \end{array}} \right]\)

If AB = C, then what is A² equal to?

Concept:

Multiplication of matrices:

• The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix.
• The result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.
• To multiply an m × n matrix by an n × p matrix, the n must be the same, and the result is an m × p matrix.

Calculation:

Given: AB = C

\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {{\rm{x}} + {\rm{y}}}&{\rm{y}}\\ {\rm{x}}&{{\rm{x}} - {\rm{y}}} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} 3\\ { - 2} \end{array}} \right] = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 4\\ { - 2} \end{array}} \right]\)

\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {3{\rm{x}} + 3{\rm{y}} - 2{\rm{y}}}\\ {3{\rm{x}} - 2{\rm{x}} + 2{\rm{y}}} \end{array}} \right] = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 4\\ { - 2} \end{array}} \right]\)

\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {3{\rm{x}} + {\rm{y}}}\\ {{\rm{x}} + 2{\rm{y}}} \end{array}} \right] = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 4\\ { - 2} \end{array}} \right]\)

So, 3x + y = 4   …. (1)

x + 2y = -2       …. (2)

Solving equation 1 and 2, we get

x = 2 and y = -2

Now,

\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {{\rm{x}} + {\rm{y}}}&{\rm{y}}\\ {\rm{x}}&{{\rm{x}} - {\rm{y}}} \end{array}} \right] \)

\(= {\rm{\;}}\left[ {\begin{array}{*{20}{c}} {2 - 2}&{ - 2}\\ 2&{2 - \left( { - 2} \right)} \end{array}} \right] = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 0&{ - 2}\\ 2&4 \end{array}} \right]\)

\({{\rm{A}}^2} = {\rm{A}} \times {\rm{A}} \)

\(= {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 0&{ - 2}\\ 2&4 \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} 0&{ - 2}\\ 2&4 \end{array}} \right]\)

\( \Rightarrow {{\rm{A}}^2} = \left[ {\begin{array}{*{20}{c}} { - 4}&{ - 8}\\ 8&{12} \end{array}} \right]\)