Determinants

Determinant

A square matrix is a matrix where the number of rows and columns are equal, that is a matrix of size n by n or n×n. The number associated with each square matrix is called the determinant of the matrix and tells us whether the matrix is invertible or not. Generally in this chapter, a matrix will mean a square matrix.

Determinant and Inverse of a 2 by 2 Matrix:
We first find the determinant of a 2 × 2 matrix and then expand to 3 × 3, .., n × n size matrices.

Example 1: Consider the general 2 × 2 matrix  and the matrix. Evaluate AB.
Solution: 

The matrix multiplication AB gives a multiple ad-bc of the identity matrix, I. This multiple (ad-bc) is called the determinant of matrix .
The determinant of a matrix A is normally denoted by det(A) or |A| and is a scalar not a matrix.
Hence the determinant of the general 2×2 matrix is defined as (2.1) det(A) = ad - bc

The determinant of a 2 × 2 matrix is the result of multiplying the entries of the leading diagonal and subtracting the product of the other diagonal. Remember the leading diagonal are the entries of the matrix which slope downwards to the right.

Example 2: Again consider the 2 × 2 matrices

Evaluate the matrix multiplication AB provided det(A) ≠ 0.
Solution:

Note that the matrix multiplication AB gives the identity matrix .
Since AB = I, what conclusions can we draw about the matrices A and B?
The given matrix  has an inverse matrix  because
we have AB = I which means B is the inverse of matrix A, that is B = A^-1.
Hence the inverse of the general 2 × 2 matrixis given by
(2.2)  provided det(A) ≠ 0

Q. What does this formula mean?
Ans: The inverse of a 2 × 2 matrix is determined by interchanging entries along the leading diagonal and placing a negative sign in the other and then multiplying this matrix by 1/det(A).

Q. What can we say if the determinant is zero, that is det(A) = 0?
Ans: If det(A) = 0 then the matrix A is non-invertible (singular), it has no inverse.

Example 3: Find the inverses of the following matrices:

Solution:
(a) Before we can find the inverse we need to evaluate the determinant. Why?
Because if the determinant is 0 then the matrix does not have an inverse. Therefore by
(2.1) 

we have

The inverse matrix A^-1 is given by the above formula (2.2) with det(A) = 13:
(b) We adopt the same procedure as part (a) to find B^-1. By
(2.1)
we have

By substituting (B)=3 into the inverse formula (2.2) we have

(c) Similarly applying (2.1) det(C) = ad-bc we have

Q. What can we conclude about the matrix C?
Ans: Since det(C) = 0 therefore the matrix C is non-invertible (singular). This means it does not have an inverse.

Properties of Determinant

Let A be a n × n matrix.

1. det(A) = det(A^T)
2. If two rows (or columns) of A are equal, then det(A) = 0.
3. If a row (or column) of A consists entirely of 0, then det(A) = 0

Example:

Let
.

Then,
 

(d) If B result from the matrix A by interchanging two rows (or columns) of A, then det(B) = -det(A).
(e) If B results from A by multiplying a row (or column) of A by a real number c, row_i(B)-c *row_i(A) (or col_i(B) = c *col_i(A)), for some i, then det(B) = c *det(A).
(f) If B results from A by adding c*row_s(A) (or c*col_s(A)) to row_r(A) (or col_r(A)), i.e., row_r(B) = row_r(A) + c*row_s(A) (or col_r(B) = col_r(A) + c*col_s(A)), then det(B) = det(A)

Example:
Let

Since B results from A by interchanging the first two rows of A,
|A| = -|B| ⇒ property (d)

Example:
Let

|B|= 2|A| ⇒ property (e),
Since col₁(B) = 2 * col₁(A)

Example:
Let

|A| = |B| ⇒ property (f),
Since row₂(B) = row₂(A) + 2 * row₁(A)
(g) If a matrix  is upper triangular (or lower triangular), then
det(A) = a₁₁a₂₂... a_nn.
(h) det(AB) = det(A) det(B)
If A is nonsingular, then
(i) det(cA) = cⁿ det(A)

Example:
Let

⇒ det(A) = 1∙2∙3 = 6 property(g)

Example:
Let

Then,

property (g)

Example:
Let

⇒ det(A) = 1∙4 - 3∙2 = -2, det(B) = 0
Thus,
det(AB) = det(A)det(B) = -2∙0 = 0 property(h)
and

Example:
Let

⇒ det(100A) = 100² det(A) = 10000(-2) = -20000
property (i)

Example:

if det(A) = -7, then
Compute:

Solution:

(j) For n × n square matrices P, Q, and X,

where I is an identity matrix.

Example:
Let

Then,

property (j)

Efficient method to compute determinant

To calculate the determinant of a complex matrix A, a more efficient method is to transform the matrix into a upper triangular matrix or a lower triangular matrix via elementary row operations. Then, the determinant of A is the product of the diagonal elements of the upper triangular matrix.
To calculate the determinant of a complex matrix A, a more efficient method is to transform the matrix into a upper triangular matrix or a lower triangular matrix via elementary row operations. Then, the determinant of A is the product of the diagonal elements of the upper triangular matrix.

Example

Note: det(A+B) is not necessarily equal to det(A) + det(B). For example,

Inverse of Matrix

The inverse (or reciprocal) of a square matrix is denoted by the A^-1, and is defined by
A × A^-1=I
For example

The 2 matrices as shown are inverses of each other, whose product is the identity matrix. Not all matrices have an inverse, and those which don't are called singular matrices.
After the previous slightly complex definitions, the calculation of the inverse matrix is relatively simple.

Clearly, if the determinant of A is zero, the inverse cannot be calculated and the matrix is said to be singular.

Inversion of 3 × 3 Matrix:
To find inverse of 3 × 3 matrix, First need to calculate determinant

Corresponding to each a_ij is a co-factor C_ij.
9 elements in 3 × 3 ⇒ 9 co-factors.
Co-factor C_ij = determinant of 2 x 2 matrix obtained by deleting row i and column j of A, prefixed by + or - according to following pattern.

Example:
C₂₃ is co-factor associated with a₂₃, in row 2 and column 3.
So delete row 2 and column 3 to give a 2 x 2 matrix

Co-factor C₂₃ is - determinant of 2X2 matrix (negative sign in position a₂₃)

Example: Find all co-factors of matrix

C₁₁ = (delete row 1 column 1, compute determinant of remaining 2X2 matrix, position a₁₁ associated with +)

C₁₂ = (delete row 1 column 2, compute determinant of remaining 2X2 matrix, position a₂₁ associated with -)

Other co-factors compute as follows:

Co-factor Matrix
Now we can find the determinant,

Multiply elements in any one row or any one column by corresponding co-factors, and sum.....
Select row 1:
|A| = a₁₁.C₁₁ + a₁₂.C₁₂ + a₁₃.C₁₃
or equivalently select column 2
|A| = a₁₂.C₁₂ + a₂₂.C₂₂ + a₃₂.C₃₂
so the determinant of

|A| = a₂₁.C₂₁ + a₂₂.C₂₂ + a₂₃.C₂₃
= (4.-11) + (3.4) + (7.6) = 10
Now we can find the Inverse......

Step 1: write matrix of co-factors

Step 2: transpose that matrix (replace rows by columns), so
Step 3: multiply each element by