If $$A=5a$$−$$b32$$ and A adj $$A=AAT$$, then $$5a+b$$ is equal to

Question

If $$A=5a$$−$$b32$$ and A adj $$A=AAT$$, then $$5a+b$$ is equal to

Infinity Learn · Accepted Answer

Given, $$__A=5a$$−$$b32$$ and A adj $$A=AAT$$ Clearly $$__A(adj$$⁡$$A)=A$$∣$$I2$$∵ if A is square matrix of order n, then $$A(adj$$⁡$$A)=(adj$$⁡$$A)$$⋅$$A=A$$∣$$InA(adj$$⁡$$A)=5a$$−$$b32I2A(adj$$⁡$$A)=(10a+3b)I2A(adj$$⁡$$A)=(10a+3b)1001A(adj$$⁡$$A)=10a+3b0010a+3b-----1$$ And $$AAT=5a$$−$$b325a3$$−$$b2AAT=25a2+b215a$$−$$2b15a$$−$$2b13$$∵$$A(adj$$⁡$$A)=AAT$$∴$$10a+3b0010a+3b=25a2+b215a$$−$$2b15a$$−$$2b13$$[ using Eqs. (i) and (ii)]⇒$$15a$$−$$2b=0$$⇒$$a=2b15$$⋯⋯⋯..$$.2$$ And $$10a+3b=13$$.$$---3On$$ substituting the value of ‘a’ from Eq.(iii) in Eq.(iv), we get $$102b15+3b=13$$⇒$$20b+45b15=13$$⇒$$65b15=13$$⇒$$b=3Now$$ , substituting the value of b in Eq. (iii) , we get $$5a=2$$  Hence, $$5a+b=2+3=5$$

$If A = [\begin{array}{cc} 5 a & - b \\ 3 & 2 \end{array}] and A adj A = A A^{T}, then 5 a + b is equal to$

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If A=5a−b32 and A adj A=AAT, then 5a+b is equal to

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$If A = [\begin{array}{cc} 5 a & - b \\ 3 & 2 \end{array}] and A adj A = A A^{T}, then 5 a + b is equal to$