Matrices

Question 1

Describe the Gaussian elimination method for solving a system of equations

Answer 1

1. Write out the equations to be solved in matrix form
2. We choose the first element in the leading diagonal as the initial point
3. Add multiples of the pivot row to the row below in order to obtain zeroes in the column of the pivot element, under the leading diagonal
4. Choose the next pivot as the next element in the leading diagonal
5. Repeat this process until the matrix is in upper triangular form, ie all elements under the leading diagonal are zero
6. Solutions can be obtained using back substitution

Question 2

Give the equation for the inverse of a 2x2 matrix

Answer 2

Question 3

Describe an adjudicate matrix, adj(A), for a 2x2 matrix

Answer 3

adj(A) is given by swapping the diagonal elements and multiplying the antidiagonal elements by -1

Question 4

State the requirement necessary for a matrix to be invertible

Answer 4

A matrix A is invertible only if det(A) ≠ 0

Question 5

Describe the transpose, T, of a matrix

Answer 5

The transpose of a matrix is such that all rows and columns are swapped