Question

If v, = (2,4) and v2 = (-1,5), then w,-V is equal to which of the following?O 18O (-2. 20)O 22O (8.-5)

Answer 1

We are given the following matrix:

`A=\begin{bmatrix}{4} & {-7} & {} \\ {-2} & {1} & {} \\ {} & {} & {}\end{bmatrix}`

We are asked to determine the coefficients of:

`A^(-1)`

Which is the inverse matrix. To do that let's remember that the inverse of a 2 by 2 matrix of the form:

`A=\begin{bmatrix}{a_1} & {a_2} & {} \\ {a_3} & {a_4} & {} \\ {} & {} & {}\end{bmatrix}`

is:

`A^(-1)=(1)/(\det A)\begin{bmatrix}{a_4} & {-a_2} & {} \\ {-a_3} & {a_1} & {} \\ {} & {} & {}\end{bmatrix}`

The value of the determinant of A (det A) is given by:

`\det A=a_1a_4-a_2a_3`

Replacing we get:

`A^(-1)=(1)/(a_1a_4-a_2a_3)\begin{bmatrix}{a_4} & {-a_2} & {} \\ {-a_3} & {a_1} & {} \\ {} & {} & {}\end{bmatrix}`

Replacing the values:

`A^(-1)=(1)/((4)(1)-(-7)(-2))\begin{bmatrix}{1_{}} & {7_{}} & {} \\ {2_{}} & {4_{}} & {} \\ {} & {} & {}\end{bmatrix}`

Solving the operations:

`A^(-1)=-(1)/(10)\begin{bmatrix}{1_{}} & {7_{}} & {} \\ {2_{}} & {4_{}} & {} \\ {} & {} & {}\end{bmatrix}`

Or:

`A^(-1)=\begin{bmatrix}{-(1)/(10)_{}} & {-(7)/(10)_{}} & {} \\ -{(1)/(5)_{}} & {-(2)/(5)_{}} & {} \\ {} & {} & {}\end{bmatrix}`

Therefore, we have:

`\begin{gathered} a=-(1)/(10) \\ b=-(7)/(10) \\ c=-(1)/(5) \\ d=-(2)/(5) \end{gathered}`