FMAT pure1 matrices and their inverses

Question 1

What is the formula for finding the inverse of a 2 × 2 matrix A?

Answer 1

For A = [[a, b], [c, d]], the inverse A⁻¹ is given by:
A⁻¹ = (1/det(A)) * [[d, -b], [-c, a]]
Where det(A) = ad - bc

Question 2

Example: Find the inverse of A where:
A = [[2, 3], [1, 4]]

Answer 2

1. Calculate det(A): det(A) = (24 - 31) = 8 - 3 = 5
2. Inverse:
   A⁻¹ = (1/5) * [[4, -3], [-1, 2]]
   = [[4/5, -3/5], [-1/5, 2/5]]

Question 3

How is the determinant of a 2 × 2 matrix calculated?

Answer 3

For A = [[a, b], [c, d]], the determinant is:
det(A) = ad - bc

Question 4

Example: Calculate the determinant of A where:
A = [[1, 2], [3, 4]].

Answer 4

det(A) = (14 - 23) = 4 - 6 = -2

Question 5

What is the minor of an element in a 3x3 matrix?

Answer 5

The minor of an element is found by eliminating the row and column of the element and calculating the determinant of the remaining 2x2 matrix.

Question 6

Example: For the matrix
2 1 4
0 3 -2
-4 1 3
Find the minors of:
a) Element 2 (row 1, column 1),
b) Element 1 (row 3, column 2),
c) Element -2 (row 2, column 3).

Answer 6

a) Minor of 2: Determinant of 2x2 matrix
3 -2
1 3
= (3)(3) - (1)(-2) = 9 + 2 = 11.

b) Minor of 1: Determinant of 2x2 matrix
2 4
0 -2
= (2)(-2) - (0)(4) = -4.

c) Minor of -2: Determinant of 2x2 matrix
2 1
-4 1
= (2)(1) - (-4)(1) = 2 + 4 = 6.

Question 7

How is the cofactor of an element in a matrix found?

Answer 7

The cofactor of an element is its minor multiplied by (-1)^(i+j), where i and j are the row and column numbers of the element.

Question 8

Example: For the matrix
2 1 4
0 3 -2
-4 1 3
Find the cofactors of:
a) Element 4 (row 1, column 3),
b) Element 0 (row 2, column 1),
c) Element -4 (row 3, column 1).

Answer 8

a) Minor of 4: Determinant of
0 3
-4 1
= (0)(1) - (-4)(3) = 12.
Cofactor: (-1)^(1+3) * 12 = 12.

b) Minor of 0: Determinant of
1 4
1 3
= (1)(3) - (4)(1) = -1.
Cofactor: (-1)^(2+1) * (-1) = 1.

c) Minor of -4: Determinant of
1 4
3 -2
= (1)(-2) - (4)(3) = -2 - 12 = -14.
Cofactor: (-1)^(3+1) * (-14) = -14.

Question 9

What is the determinant of a 3x3 matrix?

Answer 9

The determinant is found by expanding along any row or column:
det(A) = Σ(element * cofactor).

Question 10

Example: Find the determinant of the matrix
2 1 4
0 3 -2
-4 1 3
by expanding along the first row.

Answer 10

det(A) = 2(cofactor of 2) + 1(cofactor of 1) + 4(cofactor of 4).

Cofactor of 2: Minor = 11, Cofactor = 11.
Cofactor of 1: Minor = -4, Cofactor = 4.
Cofactor of 4: Minor = 12, Cofactor = 12.

det(A) = 2(11) + 1(4) + 4(12) = 22 + 4 + 48 = 74.

Question 11

What are the steps to find the inverse of a 3x3 matrix?

Answer 11

1. Find the determinant.
2. Calculate the cofactor for each element.
3. Replace each element with its cofactor to form the cofactor matrix.
4. Transpose the cofactor matrix (adjugate matrix).
5. Divide the adjugate matrix by the determinant.

Question 12

Example: Find the inverse of the matrix
2 2 0
1 4 2
2 1 1

Answer 12

1. Determinant:
   Expand along the first row:
   det = 2(det of 4 2
   1 1) - 2(det of 1 2
   2 1) + 0.
   = 2(41 - 21) - 2(11 - 22) + 0 = 4 - (-6) = 10.
2. Cofactors:
   Cofactor matrix =
   [[ 6, -5, -7],
   [-2, 2, 2],
   [ 4, -4, 6]].
3. Transpose the cofactor matrix:
   Adjugate =
   [[ 6, -2, 4],
   [-5, 2, -4],
   [-7, 2, 6]].
4. Divide by determinant:
   Inverse =
   [[ 6/10, -2/10, 4/10],
   [-5/10, 2/10, -4/10],
   [-7/10, 2/10, 6/10]].
   Simplified:
   [[ 0.6, -0.2, 0.4],
   [-0.5, 0.2, -0.4],
   [-0.7, 0.2, 0.6]]