If A = `[[3,1] , [7,5]]`, find the values of x and y such that A2 + xI2 = yA.

Given that: A = `[[3,1] , [7,5]]` Now, A2 = A.A = `[[3,1] , [7,5]] [[3,1] , [7,5]]` = `[[9 +7, 3+5], [ 21 + 35, 7 + 25]]` = `[[16,8], [56,32]]` &there4; A2 + xI2 = yA `[[16,8], [56,32]] + x [[ 1,0], [0,1]] = y [[3,1] ,[7,5]]` `[[16 +x ,8],[ 56 , 32+x]] = [[3y, y],[7y,5y]]` Comparing the corresponding entries of equal matrices, we have ⇒ y = 8 and 16 + x = 3y&there4; x = 3 x 8 - 16 = 24 - 16 = 8 Hence, the required values of x is 8 and y is 8.

If a = (3)/(7) (1)/(5), Find the Values of X and Y Such that A2 + Xi2 = Ya - Mathematics

प्रश्न

If A = $[\begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix}]$ , find the values of x and y such that A² + xI₂ = yA.

योग

उत्तर

Given that: A = $[\begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix}]$

Now, A² = A.A = $[\begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix}] [\begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix}]$

= $[\begin{matrix} 9 + 7 & 3 + 5 \\ 21 + 35 & 7 + 25 \end{matrix}]$

= $[\begin{matrix} 16 & 8 \\ 56 & 32 \end{matrix}]$

∴ A²+ xI₂ = yA

$[\begin{matrix} 16 & 8 \\ 56 & 32 \end{matrix}] + x [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = y [\begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix}]$

$[\begin{matrix} 16 + x & 8 \\ 56 & 32 + x \end{matrix}] = [\begin{matrix} 3 y & y \\ 7 y & 5 y \end{matrix}]$

Comparing the corresponding entries of equal matrices, we have ⇒ y = 8

and 16 + x = 3y
∴ x = 3 x 8 - 16
= 24 - 16
= 8

Hence, the required values of x is 8 and y is 8.

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Equality of Matrices

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2013-2014 (March)

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