Let <math> <mrow> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mrow> </math> . If M and N are two matrices given by <math> <mrow> <mi>M</mi> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mi>N</mi> <mo>=</mo> </mrow> </mstyle> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </math> then MN2 is:

A = [ 0 − 2 2 0 ] A 2 = [ 0 − 2 2 0 ] [ 0 − 2 2 0 ] = [ − 4 0 0 − 4 ] = − 4 [ 1 0 0 1 ] = − 4 l N 2 = ( 2 2 0 − 1 ) 2 2 5 l M N 2 = 4 1 2 5 ( 2 2 0 − 1 ) 3 l = [ λ 0 0 0 λ 0 0 0 λ ] , λ ≠ k

Let <math> <mrow> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mrow> </math> . If M and N are two matrices given by <math> <mrow> <mi>M</mi> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mi>N</mi> <mo>=</mo> </mrow> </mstyle> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </math> &nbsp;then MN2 is:

a non-identity symmetric matrix

neither symmetric nor skew-symmetric matrix

Ask & Answer Question

Maths NCERT Exemplar Solutions Class 11th Chapter Two Maths Class 12th Maths Matrices

Let $A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$ . If M and N are two matrices given by $M = \sum_{k = 1}^{10} A^{2 k} a n d N = \sum_{k = 1}^{10} A^{2 k - 1}$ then MN² is:

Option 1 -

a non-identity symmetric matrix

Option 2 -

a skew-symmetric matrix

Option 3 -

neither symmetric nor skew-symmetric matrix

Option 4 -

an identity matrix

2 Views | Posted 4 weeks ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

4 weeks ago

Correct Option - 4

Detailed Solution:

$A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$

$A^{2} = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$

$= [\begin{matrix} - 4 & 0 \\ 0 & - 4 \end{matrix}] = - 4 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = - 4 l$

$N^{2} = \frac{{(2^{20} - 1)}^{2}}{25} l$

$M N^{2} = \frac{4}{125} {(2^{20} - 1)}^{3} l$

$= [\begin{matrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{matrix}], λ \neq k$

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Let $A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$ . If M and N are two matrices given by $M = \sum_{k = 1}^{10} A^{2 k} a n d N = \sum_{k = 1}^{10} A^{2 k - 1}$ then MN² is:

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Let A = [ 0 − 2 2 0 ] . If M and N are two matrices given by M = ∑ k = 1 1 0 A 2 k a n d N = ∑ k = 1 1 0 A 2 k − 1 then MN2 is:

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Let $A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$ . If M and N are two matrices given by $M = \sum_{k = 1}^{10} A^{2 k} a n d N = \sum_{k = 1}^{10} A^{2 k - 1}$ then MN² is: