Determinant of a Matrix
The determinant of a matrix is a special number that can be calculated from a square matrix.
A Matrix is an array of numbers:
A Matrix
(This one has 2 Rows and 2 Columns)
The determinant of that matrix is (calculations are explained later):
3×6 − 8×4 = 18 − 32 = −14
Symbol
|A| means the determinant of the matrix A
For a 2×2 Matrix
For a 2×2 matrix (2 rows and 2 columns):
The determinant is:
|A| = ad - bc
"The determinant of A equals a times d minus b times c"
Example:
|B| = 4×8 - 6×3
= 32-18
= 14
For a 3×3 Matrix
For a 3×3 matrix (3 rows and 3 columns):
The determinant is:
|A| = a(ei - fh) - b(di - fg) + c(dh - eg)
"The determinant of A equals ... etc"
It may look complicated, but there is a pattern:
Example:
|C| = 6×(-2×7 - 5×8) - 1×(4×7 - 5×2) + 1×(4×8 - -2×2)
= 6×(-54) - 1×(18) + 1×(36)
= -306
Matrix Inverse
The inverse of a square matrix A , sometimes called a reciprocal matrix, is a matrix
such that
𝐴 𝐴−1 = 𝐼
(1)
where 𝐼 is the identity matrix. 𝐴−1 to denote the inverse matrix.
A square matrix 𝐴−1 has an inverse iff the determinant |𝐴| ≠ 0 The so-called invertible matrix
For a
matrix
(2)
the matrix inverse is
(3)
(4)
For a
matrix
(5)
the matrix inverse is
(6)
AA-1 = A-1A = I
Exampl
e:
For matrix
, its inverse is
since
AA-1 =
and A-1A =
1. Shortcut for 2x2 matrices
For
Example:
, the inverse can be found using this formula:
. Augmented matrix method
Use Gauss-Jordan elimination to transform [ A | I ]
into [ I | A-1 ].
Example: The following steps result in
so we see that
.
.
. Adjoint method
A-1 =
(adjoint of A) or A-1 =
(cofactor matrix of A)T
Example: The following steps result in A-1 for
.
The cofactor matrix for A is
so the adjoint is
Since det A = 22, we get
,
.
.
Adjoint
The matrix formed by taking the transpose of the cofactor matrix of a given
original matrix. The adjoint of matrix A is often written adj A.
Note: This is in fact only one type of adjoint. More generally, an adjoint of a
matrix is any mapping of a matrix which possesses certain properties. Consult a
book on linear algebra for more information.
Example:
Find the adjoint of the following matrix:
Solution:
First find the cofactor of each element.
As a result the cofactor matrix of A is
Finally the adjoint of A is the transpose of the cofactor matrix: