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Matrices

Short Answer Type

131. Give example to show that .
AB = O = BA where A ≠ O, B ≠ O

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132. Give example to show that .
AB = AC but B ≠ C.

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133. Give example to show that
(A + B)² = A² + B² + 2 AB.

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134.

If space straight A equals open square brackets table row 1 2 row 2 1 end table close square brackets comma space then space show space that space space space straight A squared space – space 3 space straight I space equals space 2 straight A.

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135.

If space space straight A equals open square brackets table row 2 3 row 1 cell 2 space end cell end table close square brackets comma space prove space that space straight A to the power of 1 space — space 4 straight A squared space plus space straight A space equals space straight O.

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136.

If space space space straight A equals space open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets

, verify that A² — 5A + 71 = O.

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137.

If space straight A equals space open square brackets table row 3 cell negative 5 end cell row cell negative 4 end cell 2 end table close square brackets space comma

Show that

straight A squared minus 5 straight A minus 141 equals 0

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138. $If space straight A equals space space open square brackets table row 3 cell space space space 2 end cell cell space space 0 end cell row 1 cell space space 4 end cell cell space space 0 end cell row 0 cell space space space 0 space end cell 5 end table close square brackets space comma space show space that space straight A squared space minus 7 straight A space plus space 10 space straight I subscript 3 space end subscript equals space 0$

Here space straight A equals space space open square brackets table row 3 cell space space space 2 end cell cell space space 0 end cell row 1 cell space space 4 end cell cell space space 0 end cell row 0 cell space space space 0 space end cell 5 end table close square brackets therefore space space space space straight A squared equals space open square brackets table row 3 cell space space space 2 end cell cell space space 0 end cell row 1 cell space space 4 end cell cell space space 0 end cell row 0 cell space space space 0 space end cell 5 end table close square brackets space open square brackets table row 3 cell space space space 2 end cell cell space space 0 end cell row 1 cell space space 4 end cell cell space space 0 end cell row 0 cell space space space 0 space end cell 5 end table close square brackets space space space space space space space space space space space space equals space open square brackets table row cell 9 plus 2 plus 0 end cell cell 6 plus 8 plus 0 end cell cell 0 plus 0 plus 0 end cell row cell 3 plus 4 plus 0 space space space end cell cell 2 plus 16 plus 0 end cell cell 0 plus 0 plus 0 end cell row cell 0 plus 0 plus 0 end cell cell 0 plus 0 plus 0 end cell cell space space 0 plus 0 plus 25 end cell end table close square brackets space equals space open square brackets table row 11 cell space space 14 end cell cell space space 0 end cell row 7 cell space space space 18 end cell cell space 0 end cell row 0 cell space space 0 end cell cell space 25 end cell end table close square brackets therefore space space space space space straight A squared space minus 7 straight A space plus 10 space straight I subscript 3 equals space open square brackets table row 14 14 0 row 7 8 0 row 0 0 25 end table close square brackets space space minus 7 space space open square brackets table row 3 cell space 2 end cell cell space 0 end cell row 1 cell space 4 end cell cell space 0 end cell row 0 cell space 0 space end cell 5 end table close square brackets space plus 10 space open square brackets table row 1 cell space space space space space 0 end cell cell space space space space 0 end cell row 0 cell space space space space 1 end cell cell space space space 0 end cell row 0 cell space space space space 0 end cell cell space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell 11 minus 21 plus 10 end cell cell 14 minus 14 plus 0 end cell cell 0 plus 0 plus 0 end cell row cell 7 minus 7 plus 0 end cell cell 18 minus 28 plus 10 end cell cell 0 plus 0 plus 0 end cell row cell 0 plus 0 plus 0 end cell cell 0 plus 0 plus 0 end cell cell 25 minus 35 plus 10 end cell end table close square brackets space equals space open square brackets table row 0 cell space space space space 0 end cell cell space space space 0 end cell row 0 cell space space space space 0 end cell cell space space space 0 end cell row 0 cell space space space space 0 end cell cell space space space 0 end cell end table close square brackets therefore space space space space straight A squared minus 7 straight A plus 10 space straight I subscript 3 space equals straight O

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139.

Find space straight A squared space minus 5 straight A plus 6 space straight I space If space straight A space equals space open square brackets table row 2 0 1 row 2 1 3 row 1 cell negative 1 end cell 0 end table close square brackets

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140.

If space straight A equals space open square brackets table row 1 2 row 2 1 end table close square brackets space comma space straight f space left parenthesis straight x right parenthesis space equals straight x to the power of 2 space end exponent minus 2 straight x minus 3 space show space that space straight f left parenthesis straight A right parenthesis equals straight O

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