Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Let A = $open square brackets table row 4 cell space space space minus 6 end cell cell space space space space 1 end cell row cell negative 1 end cell cell space minus 1 end cell cell space space 1 end cell row cell negative 4 end cell cell space space space 11 end cell cell negative 1 end cell end table close square brackets.$
Show that A is inevitable. Find adj. A and A Also verify that AA^–1 = A^–1 A = I.

72 Views

212.

Verify AA^–1 = A^–1 A = I where
$straight A space equals open square brackets table row 0 cell space space space 0 end cell cell space space minus 1 end cell row 3 cell space space space space 4 end cell cell space space space space space 5 end cell row cell negative 2 end cell cell negative 4 end cell cell space space minus 7 end cell end table close square brackets.$

79 Views

Short Answer Type

213.

If $straight A space equals space open square brackets table row 3 cell space space minus 3 end cell cell space space space 4 end cell row 2 cell space space minus 3 end cell cell space space 4 end cell row 0 cell space minus 1 end cell cell space space 1 end cell end table close square brackets.$ then A³ = A^–1What is adj. A ?

91 Views

Long Answer Type

214.

If $straight A space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space minus 1 end cell cell space space 0 end cell row 1 cell space space space space 0 end cell cell space space 0 end cell end table close square brackets comma$ find A^–1 and show that A^–1 = A².

72 Views

215.

If $straight A space equals space open square brackets table row 3 cell space space minus 3 end cell cell space space space 4 end cell row 2 cell space space minus 3 end cell cell space space space 4 end cell row 0 cell space space minus 1 end cell cell space space space 1 end cell end table close square brackets comma$ show that A⁴ = I. Hence find A^–1

76 Views

216.
If A = $open square brackets table row 2 cell space space 0 end cell cell space space minus 1 end cell row 5 cell space space 1 end cell cell space space space space 0 end cell row 0 cell space space 1 end cell cell space space space space 3 end cell end table close square brackets comma$ prove that A^–1 = A^–2 – 6 A + 111.

Here $straight A space equals space open square brackets table row 2 cell space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space 0 end cell row 0 cell space space space 1 end cell cell space space space space 3 end cell end table close square brackets$

$therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space 0 end cell cell negative 1 end cell row 5 cell space space 1 end cell cell space space space 0 end cell row 0 cell space space 1 end cell cell space space 3 end cell end table close vertical bar space equals space 2 space open vertical bar table row 1 cell space space space 0 end cell row 1 cell space space 3 end cell end table close vertical bar space minus space 0 space open vertical bar table row 5 cell space space 0 end cell row 0 cell space space 3 end cell end table close vertical bar space plus space left parenthesis negative 1 right parenthesis space open vertical bar table row 5 cell space space 1 end cell row 0 cell space space 1 end cell end table close vertical bar space space space equals space 2 left parenthesis 3 minus 0 right parenthesis minus 0 minus left parenthesis 5 minus 0 right parenthesis space equals space 6 minus 5 space equals space 1 space not equal to space 0 therefore space space space straight A to the power of negative 1 end exponent space exists$
Co-factors of the elements of first row of | A | are
$open vertical bar table row 1 cell space space space 0 end cell row 1 cell space space space 3 end cell end table close vertical bar comma space space space minus open vertical bar table row 5 cell space space space 0 end cell row 0 cell space space space 3 end cell end table close vertical bar comma space space space space space open vertical bar table row 5 cell space space space 1 end cell row 0 cell space space space 1 end cell end table close vertical bar$
i.e. 3, – 15, 5 respectively
Co-factors of the elements of second row of | A | are
$negative open vertical bar table row 0 cell space space space minus 1 end cell row 1 cell space space space space space 3 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space space minus 1 end cell row 0 cell space space space space space space 3 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space space space 0 end cell row 0 cell space space space space space 1 end cell end table close vertical bar$
i.e. – 1, 6, – 2 respectively
Co-factors of the elements of third row of | A | are
$open vertical bar table row 0 cell space space space minus 1 end cell row 1 cell space space space space space 0 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space minus 1 end cell row 5 cell space space space space space 0 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space space space 0 end cell row 5 cell space space space space space 1 end cell end table close vertical bar$

i.e. 1, – 5, 2 respectively
$therefore space space space adj. space straight A space equals space open square brackets table row cell space space 3 end cell cell space space space minus 15 end cell cell space space space space 5 end cell row cell negative 1 end cell cell space space space space space 6 end cell cell negative 2 end cell row 1 cell space space space minus 5 end cell cell space space 2 end cell end table close square brackets to the power of apostrophe space equals space space open square brackets table row cell space space 3 end cell cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space minus 5 end cell row cell space space 5 end cell cell negative 2 end cell cell space space space space space 2 end cell end table close square brackets space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 over 1 open square brackets table row cell space space space space 3 end cell cell space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space space 6 end cell cell space space minus 5 end cell row cell space space space space space 5 end cell cell space space space minus 2 end cell cell space space space space space 2 end cell end table close square brackets space equals space open square brackets table row cell space space space space 3 end cell cell space space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space space 6 end cell cell space space minus 5 end cell row cell space space space space space 5 end cell cell space space space minus 2 end cell cell space space space space space 2 end cell end table close square brackets straight A squared space equals space open square brackets table row 2 cell space space space space 0 end cell cell space space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space space space 1 end cell cell space space space space space space 3 end cell end table close square brackets space open square brackets table row 2 cell space space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space 1 end cell cell space space space 3 end cell end table close square brackets space space space equals space open square brackets table row cell 4 plus 0 plus 0 end cell cell space space space 0 plus 0 minus 1 end cell cell space space minus 2 plus 0 minus 3 end cell row cell 10 plus 5 plus 0 end cell cell 0 plus 1 plus 0 end cell cell negative 5 plus 0 plus 0 end cell row cell 0 plus 5 plus 0 end cell cell 0 plus 1 plus 3 end cell cell 0 plus 0 plus 9 end cell end table close square brackets space equals space open square brackets table row 4 cell space space minus 1 end cell cell space space space minus 5 end cell row 15 cell space space space space 1 end cell cell space space minus 5 end cell row 5 cell space space space 4 end cell cell space space space 9 end cell end table close square brackets straight R. straight H. straight S space equals space straight A squared minus 6 straight A space plus space 11 space straight I space space space space space space space space space space space space space space space$
$equals space open square brackets table row 4 cell space space minus 1 end cell cell space space space minus 5 end cell row 15 cell space space space space space 1 end cell cell space space minus 5 end cell row 5 cell space space space space 4 end cell cell space space space space space 9 end cell end table close square brackets minus 6 space open square brackets table row 2 cell space space space 0 end cell cell space space minus 1 end cell row 5 cell space space space 1 end cell cell space space space space space 0 end cell row 0 cell space space space 1 end cell cell space space space space space 3 end cell end table close square brackets space plus space 11 space open square brackets table row 1 cell space space space 0 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 0 cell space space 0 end cell cell space space 1 end cell end table close square brackets space equals space open square brackets table row 4 cell space space minus 1 end cell cell space space minus 5 end cell row 15 cell space space space space space 1 end cell cell space minus 5 end cell row 5 cell space space space 4 end cell cell space space space space 9 end cell end table close square brackets space plus space open square brackets table row cell negative 12 end cell cell space space space space 0 end cell cell space space space 6 end cell row cell negative 30 end cell cell space minus 6 end cell cell space space space 0 end cell row 0 cell space space minus 6 end cell cell space space space minus 18 end cell end table close square brackets space plus space space open square brackets table row 11 cell space 0 end cell cell space space 0 end cell row 0 cell space space 11 end cell cell space space 0 end cell row 0 0 cell space space 11 end cell end table close square brackets equals space open square brackets table row cell 4 minus 12 plus 11 end cell cell space space space minus 1 plus 0 plus 0 end cell cell space space space space space minus 5 plus 6 plus 0 end cell row cell 15 minus 30 plus 0 end cell cell space space 1 minus 6 plus 11 end cell cell space space minus 5 plus 0 plus 0 end cell row cell 5 plus 0 plus 0 end cell cell space 4 minus 6 plus 0 end cell cell space space space space space 9 minus 18 plus 11 end cell end table close square brackets space equals space open square brackets table row cell space space 3 end cell cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space minus 5 end cell row cell space space space 5 end cell cell space minus 2 end cell cell space space space space 2 end cell end table close square brackets equals space straight A squared space equals space straight R. straight H. straight S.$

74 Views

217.

If $straight A space equals space open square brackets table row 1 cell space space 2 end cell cell space space 2 end cell row 2 cell space space 1 end cell cell space space 2 end cell row 2 cell space space 2 end cell cell space space 1 end cell end table close square brackets comma space$ Find A^–1 and hence prove that A² – 4A – 5 I = O.

76 Views

218.

For the matrix $straight A space equals space open square brackets table row 1 cell space space 1 end cell cell space space space space space 1 end cell row 1 cell space space space 2 end cell cell space minus 3 end cell row 2 cell negative 1 end cell cell space space space space 3 end cell end table close square brackets comma$ show that A³ + 6A² + 5A + 11 I = 0. Hence, find A^–1.

89 Views

219.

If $straight A space equals space open square brackets table row cell space space 2 end cell cell space space space minus 1 end cell cell space space space space space 1 end cell row cell negative 1 end cell cell space space space space space 2 end cell cell space minus 1 end cell row cell space space 1 end cell cell space space minus 1 end cell cell space space space space 2 end cell end table close square brackets comma$
verify that A¹ – 6A² + 9A – 4 I = O and hence find A^–1.

74 Views

220.

Compute (AB)^-1 where $straight A space equals space open square brackets table row 5 cell space space space 0 end cell cell space space 4 end cell row 2 cell space space 3 space end cell cell space 2 end cell row 1 cell space 2 end cell cell space 1 end cell end table close square brackets comma space space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space 2 end cell cell space space 3 end cell row 1 cell space space 4 end cell cell space space 3 end cell row 1 cell space space 3 end cell cell space space 4 end cell end table close square brackets$

73 Views

Previous Year Papers

Test Series

Test Yourself

Long Answer Type

Short Answer Type

Long Answer Type

216. If A = prove that A–1 = A–2 – 6 A + 111.