If A = and A adj. A = A AT, then 5a + b is equal to

Engineering

Mathematics

Transpose and Adjoint of a Matrix

Question

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

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A adj. A =A A^T

| A | I_n = A A^T

(10a + 3b) I_n = $[\begin{matrix} 5 a & - b \\ 3 & 2 \end{matrix}] [\begin{matrix} 5 a & 3 \\ - b & 2 \end{matrix}] = [\begin{matrix} 25 a^{2} + b^{2} & 15 a - 2 b \\ 15 a - 2 b & 9 + 4 \end{matrix}]$

10a + 3b = 13, 10a + 3b = 25a² + b²,

15a – 2b = 0 ⇒ a = $\frac{2 b}{15}$

10 × $\frac{2 b}{15}$ + 3b = 13

4b + 9b = 39 ⇒ b = 3, a = $\frac{2}{5}$

5a + b = 5.