Determinants | Engineering Mathematics - Civil Engineering (CE) PDF Download

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 Determinants | Engineering Mathematics - Civil Engineering (CE) and the matrixDeterminants | Engineering Mathematics - Civil Engineering (CE). Evaluate AB.
Solution: 
Determinants | Engineering Mathematics - Civil Engineering (CE)
The matrix multiplication AB gives a multiple ad-bc of the identity matrix, I. This multiple (ad-bc) is called the determinant of matrix Determinants | Engineering Mathematics - Civil Engineering (CE).
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 Determinants | Engineering Mathematics - Civil Engineering (CE)is defined as (2.1) det(A) = ad - bc
Determinants | Engineering Mathematics - Civil Engineering (CE)
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
Determinants | Engineering Mathematics - Civil Engineering (CE)
Evaluate the matrix multiplication AB provided det(A) ≠ 0.
Solution:
Determinants | Engineering Mathematics - Civil Engineering (CE)
Note that the matrix multiplication AB gives the identity matrix Determinants | Engineering Mathematics - Civil Engineering (CE).
Since AB = I, what conclusions can we draw about the matrices A and B?
The given matrix Determinants | Engineering Mathematics - Civil Engineering (CE) has an inverse matrix Determinants | Engineering Mathematics - Civil Engineering (CE) 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 matrixDeterminants | Engineering Mathematics - Civil Engineering (CE)is given by
(2.2) Determinants | Engineering Mathematics - Civil Engineering (CE) 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:

Determinants | Engineering Mathematics - Civil Engineering (CE)
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) Determinants | Engineering Mathematics - Civil Engineering (CE)

Determinants | Engineering Mathematics - Civil Engineering (CE)
we have
Determinants | Engineering Mathematics - Civil Engineering (CE)
The inverse matrix A-1 is given by the above formula (2.2) with det(A) = 13:Determinants | Engineering Mathematics - Civil Engineering (CE)
(b) We adopt the same procedure as part (a) to find B-1. By
(2.1) Determinants | Engineering Mathematics - Civil Engineering (CE)Determinants | Engineering Mathematics - Civil Engineering (CE)
we have
Determinants | Engineering Mathematics - Civil Engineering (CE)
By substituting (B)=3 into the inverse formula (2.2) we have
Determinants | Engineering Mathematics - Civil Engineering (CE)
(c) Similarly applying (2.1) det(C) = ad-bc we have
Determinants | Engineering Mathematics - Civil Engineering (CE)


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(AT)
  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
Determinants | Engineering Mathematics - Civil Engineering (CE).

Then,
 Determinants | Engineering Mathematics - Civil Engineering (CE)Determinants | Engineering Mathematics - Civil Engineering (CE)
Determinants | Engineering Mathematics - Civil Engineering (CE)
(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, rowi(B)-c *rowi(A) (or coli(B) = c *coli(A)), for some i, then det(B) = c *det(A).
(f) If B results from A by adding c*rows(A) (or c*cols(A)) to rowr(A) (or colr(A)), i.e., rowr(B) = rowr(A) + c*rows(A) (or colr(B) = colr(A) + c*cols(A)), then det(B) = det(A)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
Since B results from A by interchanging the first two rows of A,
|A| = -|B| ⇒ property (d)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
|B|= 2|A| ⇒ property (e),
Since col1(B) = 2 * col1(A)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
|A| = |B| ⇒ property (f),
Since row2(B) = row2(A) + 2 * row1(A)
(g) If a matrix Determinants | Engineering Mathematics - Civil Engineering (CE) is upper triangular (or lower triangular), then
det(A) = a11a22… ann.
(h) det(AB) = det(A) det(B)
If A is nonsingular, then Determinants | Engineering Mathematics - Civil Engineering (CE)
(i) det(cA) = cn det(A)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
⇒ det(A) = 1∙2∙3 = 6 property(g)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
Then,
Determinants | Engineering Mathematics - Civil Engineering (CE)
property (g)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
⇒ det(A) = 1∙4 - 3∙2 = -2, det(B) = 0
Thus,
det(AB) = det(A)det(B) = -2∙0 = 0 property(h)
and
Determinants | Engineering Mathematics - Civil Engineering (CE)

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
⇒ det(100A) = 1002 det(A) = 10000(-2) = -20000
property (i)

Example:
Determinants | Engineering Mathematics - Civil Engineering (CE)
if det(A) = -7, then
Compute:
Determinants | Engineering Mathematics - Civil Engineering (CE)

Solution:
Determinants | Engineering Mathematics - Civil Engineering (CE)
(j) For n × n square matrices P, Q, and X,
Determinants | Engineering Mathematics - Civil Engineering (CE)
where is an identity matrix.

Example:
Let
Determinants | Engineering Mathematics - Civil Engineering (CE)
Then,
Determinants | Engineering Mathematics - Civil Engineering (CE)
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
Determinants | Engineering Mathematics - Civil Engineering (CE)
Determinants | Engineering Mathematics - Civil Engineering (CE)

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

Determinants | Engineering Mathematics - Civil Engineering (CE)

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
Determinants | Engineering Mathematics - Civil Engineering (CE)
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.
Determinants | Engineering Mathematics - Civil Engineering (CE)
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
Determinants | Engineering Mathematics - Civil Engineering (CE)
Corresponding to each aij is a co-factor Cij.
9 elements in 3 × 3 ⇒ 9 co-factors.
Co-factor Cij = determinant of 2 x 2 matrix obtained by deleting row i and column j of A, prefixed by + or – according to following pattern.
Determinants | Engineering Mathematics - Civil Engineering (CE)

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
Determinants | Engineering Mathematics - Civil Engineering (CE)
Co-factor C23 is – determinant of 2X2 matrix (negative sign in position a23)
Determinants | Engineering Mathematics - Civil Engineering (CE)

Example: Find all co-factors of matrix
Determinants | Engineering Mathematics - Civil Engineering (CE)
C11 = (delete row 1 column 1, compute determinant of remaining 2X2 matrix, position a11 associated with +)
Determinants | Engineering Mathematics - Civil Engineering (CE)
C12 = (delete row 1 column 2, compute determinant of remaining 2X2 matrix, position a21 associated with -)
Determinants | Engineering Mathematics - Civil Engineering (CE)
Other co-factors compute as follows:
Determinants | Engineering Mathematics - Civil Engineering (CE)
Co-factor Matrix Determinants | Engineering Mathematics - Civil Engineering (CE)
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
Determinants | Engineering Mathematics - Civil Engineering (CE)
|A| = a21.C21 + a22.C22 + a23.C23
= (4.-11) + (3.4) + (7.6) = 10
Now we can find the Inverse……
Determinants | Engineering Mathematics - Civil Engineering (CE)
Step 1: write matrix of co-factors
Determinants | Engineering Mathematics - Civil Engineering (CE)
Step 2: transpose that matrix (replace rows by columns), soDeterminants | Engineering Mathematics - Civil Engineering (CE)
Step 3: multiply each element by Determinants | Engineering Mathematics - Civil Engineering (CE)
Determinants | Engineering Mathematics - Civil Engineering (CE)
Determinants | Engineering Mathematics - Civil Engineering (CE)

The document Determinants | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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FAQs on Determinants - Engineering Mathematics - Civil Engineering (CE)

1. What is the determinant of a matrix?
Ans. The determinant of a matrix is a scalar value that can be calculated from the elements of a square matrix and provides important information about the matrix's properties.
2. What are some properties of determinants?
Ans. Some properties of determinants include linearity in rows or columns, changing rows or columns changes the sign of the determinant, and the determinant of the identity matrix is 1.
3. What is an efficient method to compute the determinant of a matrix?
Ans. One efficient method to compute the determinant of a matrix is using cofactor expansion along a row or column, also known as Laplace's expansion.
4. How can determinants be used to find the inverse of a matrix?
Ans. The inverse of a matrix can be found by using the formula A^(-1) = (1/det(A)) * adj(A), where det(A) is the determinant of matrix A and adj(A) is the adjugate of matrix A.
5. What are some common applications of determinants in mathematics?
Ans. Determinants are commonly used in solving systems of linear equations, finding the area of a parallelogram or triangle, and in calculating eigenvalues and eigenvectors of a matrix.
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