Exercise Set 4
Matrices and Determinants

1. Consider a 2 x 2 matrix, A. The elements of the matrix are as follows: a₁₁ = 15, a₁₂ = -2, a₂₁ = 30, a₂₂ = 4.

Obtain the following:

1. The determinant of A.
2. The minors of a₁₁, a₁₂, a₂₁ and a₂₂.
3. The cofactors of a₁₁, a₁₂, a₂₁ and a₂₂.
4. The cofactor matrix, C.
5. The adjoint of A. Denote this by J.
6. The inverse of A. Denote this by K.
7. What is the product of A and A^-1?

2. Define a matrix, B, with elements as follows: b₁₁ = 3, b₁₂ = 0, b₂₁ = 4, b₂₂ = 6.

1. Obtain E as the sum of A and B.
2. Obtain F as the product of A and B.
3. Obtain G as the product of B and A. Are F and G the same?
4. Obtain B^-1. Confirm that BB^-1 = I.

3. Define a 3 x 3 matrix, S. Select values for elements in each row and column, nine in all.

Obtain the following:

1. The determinant of S. Is the matrix singular? If yes, revise your values.
2. The minors of s₁₁, s₁₂, s₁₃, s₂₁, s₂₂, s₂₃, s₃₁, s₃₂ and s₃₃.
3. The cofactors of s₁₁, s₁₂, s₁₃, s₂₁, s₂₂, s₂₃, s₃₁, s₃₂ and s₃₃.
4. The cofactor matrix, C.
5. The adjoint of S: adj(S)
6. The inverse of S: S^-1
7. What is the product of S and S^-1?

1. Obtain M = ST.
2. Is it possible to multiply T by S? Explain.