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(B) Given $A = \begin{bmatrix} a & -b \ b & a \end{bmatrix}$ and $ab = 5/2 \quad \dots (i)$ The transpose $A^T = \begin{bmatrix} a & b \ -b & a \end

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$2x^2 \pm 10x + 5 = 0$

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(B) Given $A = \begin{bmatrix} a & -b \ b & a \end{bmatrix}$ and $ab = 5/2 \quad \dots (i)$ The transpose $A^T = \begin{bmatrix} a & b \ -b & a \end{bmatrix}$. Then $AA^T = \begin{bmatrix} a & -b \ b & a \end{bmatrix} \begin{bmatrix} a & b \ -b & a \end{bmatrix} = \begin{bmatrix} a^2 + b^2 & 0 \ 0 & a^2 + b^2 \end{bmatrix}$. Given $AA^T = 20I = \begin{bmatrix} 20 & 0 \ 0 & 20 \end{bmatrix}$,we have $a^2 + b^2 = 20$. Using the identity $(a+b)^2 = a^2 + b^2 + 2ab$,we get $(a+b)^2 = 20 + 2(5/2) = 20 + 5 = 25$. Thus,$a+b = \pm 5$. The quadratic equation with roots $a$ and $b$ is $x^2 - (a+b)x + ab = 0$. Substituting the values,$x^2 \mp 5x + 5/2 = 0$. Multiplying by $2$,we get $2x^2 \mp 10x + 5 = 0$.

Let $a$ and $b$ be non-zero real numbers such that $ab = 5/2$. Given $A = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and $AA^T = 20I$ (where $I$ is the identity matrix),the quadratic equation whose roots are $a$ and $b$ is:

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