If A = 5 a - b 3 2 and A . a d j A = A A T , then 5 a + b is equal to

A = 5 a - b 3 2 and A T = 5 a 3 - b 2 A A T = 25 a 2 + b 2 15 a - 2 b 15 a - 2 b 13 Now, A . a d j A = A I 2 = 10 a + 3 b 0 0 10 a + 3 b Given A A T = A . a d j A 15 a - 2 b = 0 ......(i) 10 a + 3 b = 13 .......(ii) Solving we get 5 a = 2 and b = 3 ∴ 5 a + b = 5

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Mathematics - Matrices Question with Solution | TestHub

MathematicsMatricesAdjointMedium2 minPYQ_2016

MathematicsMediumsingle choice

If $A = [\begin{matrix} 5 a & - b \\ 3 & 2 \end{matrix}]$ and $A . a d j A = A A^{T}$ , then $5 a + b$ is equal to

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Answer:

Solution:

$A = [\begin{matrix} 5 a & - b \\ 3 & 2 \end{matrix}]$ and $A^{T} = [\begin{matrix} 5 a & 3 \\ - b & 2 \end{matrix}]$

$A A^{T} = [\begin{matrix} 25 a^{2} + b^{2} & 15 a - 2 b \\ 15 a - 2 b & 13 \end{matrix}]$

Now, $A . a d j A = |A| I_{2} = [\begin{matrix} 10 a + 3 b & 0 \\ 0 & 10 a + 3 b \end{matrix}]$

Given $A A^{T} = A . a d j A$

$15 a - 2 b = 0$ ......(i)

$10 a + 3 b = 13$ .......(ii)

Solving we get

$5 a = 2$ and $b = 3$

$∴$ $5 a + b = 5$

Stream:JEESubject:MathematicsTopic:MatricesSubtopic:Adjoint

⏱ 2mℹ️ Source: PYQ_2016

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