Finding the Inverse of the Coefficient Matrix
To find the inverse of a 2* 2 matrix, we use the corresponding formula.
| Matrix
|
Inverse
|
| A= [ cc a & b c & d ]
|
A^(- 1)=1/ad-bc [ cc d & - b - c & a ] where ad-bc ≠0
|
The expression ad-bc is known as the
determinant of a 2* 2 matrix. Because it is in the denominator of a fraction, if the determinant is zero, the matrix cannot have an inverse. Consider our coefficient matrix.
[
cc
4 & 3 2 & 2
]
Let's calculate its determinant.
Since the determinant is not zero, the matrix has an inverse. We can now apply the formula for the inverse. Note that we usually refer to the determinant using the notation ad-bc=det(A).
1/det(A) d & - b - c & a
1/2 2 & - 3 - 2 & 4
1/2 2 & - 3 - 2 & 4
For simplicity, instead of multiplying the matrix by the scalar, we will leave the expression as it is.
Multiplying Matrices
In our exercise, we had to multiply a 2* 2 matrix by a 2* 1 matrix. This is possible because the number of columns of the first matrix is equal to the number of rows of the second one. Moreover, the product has as many rows as the first matrix and as many columns as the second one. Thus, the product is a 2* 1 matrix.
[
cc
a & b
c & d
]
*
[
c
x
y
]
=
[
c
ax+by
cx+dy
]
2 * 2 2 * 1 2 * 1
Let's multiply the matrices of the exercise.
2 & - 3 - 2 & 4 * 21 9
2(21)+(- 3)(9) - 2(21)+4(9)
42-27 - 42+36
15 - 6
