Question

if A = 3 2 and B = a b find a b such that
4 1 3 5

AB = BA. Compute 3 A +5B.

Answer 1

Answer on Question#39416, Math, Matrix

Let us multiply $AB$ and $BA$. $AB = \begin{pmatrix} 3a + 6 & 3b + 10 \\ 4a + 3 & 4b + 5 \end{pmatrix}$, $BA = \begin{pmatrix} 3a + 4b & 2a + b \\ 29 & 11 \end{pmatrix}$. Using the second row of identity $AB = BA$, obtain $4a + 3 = 29; 4b + 5 = 11$, thus $a = 6; b = \frac{3}{2}$.

$$A = \begin{pmatrix} 3 & 2 \\ 4 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 6 & \frac{3}{2} \\ 3 & 5 \end{pmatrix}.$$ $$3A + 5B = \begin{pmatrix} 39 & 13.5 \\ 27 & 28 \end{pmatrix}.$$