How do you know if the inverse of 2x2 matrix exsists or not? Can't you just use the formula (A^-1) for each matrix?(5,0_0,1), (1,2_2,1), (6,3_8,4), (-3,-2_6,4) <-- which one of these matrices...

How do you know if the inverse of 2x2 matrix exsists or not? Can't you just use the formula (A^-1) for each matrix?

(5,0_0,1), (1,2_2,1), (6,3_8,4), (-3,-2_6,4) <-- which one of these matrices have inverses that do not exist?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to calculate the determinant of the matrix. If the value of determinant is other than zero, then the inverse of the matrix exists.

Evaluating the determinant of the matrix `((5,0),(0,1))` yields:

`Delta =`  `[[5,0],[0,1]]` `= 5*1 - 0*0 = 5 != 0`

Notice that the determinant of this matrix is not zero, hence the inverse of the matrix exists.

Evaluating the determinant of the matrix `((1,2),(2,1))` yields:

`Delta =`  `[[1,2],[2,1]]` `= 1*1 - 2*2 = 1 - 4 = -3!=0`

Notice that the determinant of this matrix is not zero, hence the inverse of the matrix exists.

Evaluating the determinant of the matrix `((6,3),(8,4))`  yields:

`Delta = [[6,3],[8,4]] = 6*4 - 8*3 = 24 - 24 = 0`

The inverse of the matrix `((6,3),(8,4))`  does not exist.

Evaluating the determinant of the matrix `((-3,-2),(6,4))`  yields:

`Delta =`  `[[-3,-2],[6,4]]` = `-3*4 - 6*(-2) = -12 + 12 = 0`

The inverse of the matrix `((-3,-2),(6,4))`  does not exist.

Hence, the inverses of matrices `((6,3),(8,4))` and `((-3,-2),(6,4))` do not exist.