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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.

space Here space straight A equals space open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets space space space space space space space space space space space straight A squared equals open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets space space space open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets space equals space open square brackets table row cell 9 minus 1 end cell cell 3 plus 2 end cell row cell negative 3 minus 2 end cell cell negative 1 plus 4 end cell end table close square brackets equals space open square brackets table row 8 5 row cell negative 5 end cell 3 end table close square brackets space space space space straight L. straight H. straight S space equals space straight A squared space minus space 5 straight A space plus 71 space equals space open square brackets table row 8 5 row cell negative 5 end cell 3 end table close square brackets space minus 5 space open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets space plus 7 space open square brackets table row 1 0 row 0 1 end table close square brackets space space space space space space space space space space space space space space space equals space open square brackets table row 8 5 row cell negative 5 end cell 3 end table close square brackets space plus space open square brackets table row 15 cell negative 5 end cell row 5 cell negative 10 end cell end table close square brackets plus space open square brackets table row 7 0 row 0 7 end table close square brackets space space space space space space space space space space space space space space space equals space open square brackets table row cell 8 minus 15 plus 7 end cell cell 5 minus 5 plus 0 end cell row cell negative 5 plus 5 plus 0 end cell cell 3 minus 10 plus 7 end cell end table close square brackets space space equals space open square brackets table row 0 0 row 0 0 end table close square brackets equals straight O space space space space space space space space space space space space space space equals space space straight R. straight H. straight S

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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

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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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Short Answer Type

136. , verify that A2 — 5A + 71 = O.

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.