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`