1MA01: Mathematical Methods Tutorial Sheet 4 1
1.
Matrix inverses using Gauss-Jordan elimination
Determine the inverse of the matrices:
A =
2 3
3 5
!
; B =
3 4
5 7
!

; C =


− 2 2 1
1 2 0
0 − 1 0



(1)
2.
Solving systems of equations with matrix inverses
Use the inverse of the matrix A in Question 1 to solve the following 2 different systems of linear equations by matrix multiplication.
2 x + 3 y = 10 a nd 2 x + 3 y = 4
3 x + 5 y = 15 3 x + 5 y = 7
Hint: first express these systems of equations in matrix form.
3.
Determinants and invertibility
Compute the determinant of the following matrices and comment on whether or not they are invertible.
A =
2 1
3 6
!
; B =
2 3
6 9
!

; C =


3 1 6
− 5 0 − 2
4 6 − 1



(2)
1
4.
Properties of matrix inverse
Verify that
• ( AB )
− 1
= B
− 1
A
− 1
• det ( AB ) = det ( A ) det ( B ) for the matrices A and B in question 1 above.