Question

If F(x,y,z)=(2x + y, x + z, y - z) and A is the matrix of F in the canonical basis, then a^2+b^2 equals to

Answer 1

Given:

$\begin{gathered} f(x,y,z)=(2x+y,x+z,y-z) \\ A=\begin{bmatrix}{a} & {1} & {0} \\ {1} & {0} & {1} \\ {0} & {b} & {-1}\end{bmatrix} \end{gathered}$

find:

$a^2+b^2$

Explanation: The matrix F can be written as

$\begin{bmatrix}{2} & {1} & {0} \\ {1} & {0} & {1} \\ {0} & {1} & {-1}\end{bmatrix}$

compare matrix F to matrix A we get,

$a=2,b=1$

so

$\begin{gathered} a^2+b^2=(2)^2+(1)^2 \\ =4+1 \\ =5 \end{gathered}$

Final answer:

$a^2+b^2=5$