Determinant of Matrices - MathsTips.com

The determinant of a matrix could be a special number that may be calculated from a square matrix. Determinants are like matrices, however, done up in absolute-value bars rather than square brackets. The determinant of a matrix could be a scalar property of the matrix. Only sq. matrices have determinants. If there is a matrix A then its determinant is written by taking numbers of elements and putting them within absolute-value bars rather than sq. brackets. The determinant is viewed as a result whose input could be a matrix and whose output could be a single number.

Uses of Determinant

Determinants are useful as a result of they tell us whether a matrix is inverted or not.
It plays a vital role in solving the linear equation.
The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
Determinant has wide application in engineering, science, economics, social science, etc.

To every square matrix $A=[a_{ij}]$ of order $a_{n}$ , we can associate a number [real or complex] called determinant of the square matrix A, where $a_{ij}=(i,j)^{th}$ element of A. It is denoted by |A| or det A.

If $A=\begin {bmatrix} a &b \\ c&d \end {bmatrix}$ then the determinant of A is written as $|A|=\begin {vmatrix} a &b \\ c&d \end {vmatrix}$ = det (A).

Note

For matrix A, |A| is read as the determinant of A and not modulus of A.
Only square matrices have determinants.

Determinant of a 2*2 matrix

Suppose A is a matrix of order 2*2 matrix such as : $A=\begin {bmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end {bmatrix}$ the the determinant of matrix A would be |A| = $\begin {vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end {vmatrix}$ = $a_{11}a_{22} - a_{21}a_{12}$

Example 1: Find the determinant of matrix $A=\begin {bmatrix} 4 &2 \\ 3&5 \end {bmatrix}$

Solution: $|A|=\begin {vmatrix} 4 &2 \\ 3&5 \end {vmatrix}$ = 4*5 – 3*2 = 20-6 = 14

Example 2: Find the determinant of matrix $A=\begin {bmatrix} 1&2 \\ 5&-3 \end {bmatrix}$

Solution: $|A|=\begin {vmatrix} 1 &2 \\ 5&-3 \end {vmatrix}$ = 1*(-3) – 5*2 = -3 – 10 = -13

Determinant of a 3*3 matrix

Suppose A is a matrix of order 3*3 matrix such as $A=\begin {bmatrix} a_{1} &b_{1} & c_{1} \\ a_{2}&b_{2}& c_{2} \\ a_{3} & b_{3}& c_{3} \end {bmatrix}$ then the determinant of matrix A would be $|A|=\begin {vmatrix} a_{1} &b_{1} & c_{1} \\ a_{2}&b_{2}& c_{2} \\ a_{3} & b_{3}& c_{3} \end {vmatrix}$ = $a_{1}\begin {vmatrix} b_{2} &c_{2} \\ b_{3}&c_{3} \end {vmatrix}$ – $b_{1}\begin {vmatrix} a_{2} &c_{2} \\ a_{3}&c_{3} \end {vmatrix}$ + $c_{1}\begin {vmatrix} a_{2} &b_{2} \\ a_{3}&b_{3} \end {vmatrix}$

The element a1,b1,c1 of first row are in the expression with alternatively positive and negative sign and each element is multiplied by a each element is multiplied by a certain determinant of order 2.

Example 3: Let $A=\begin {bmatrix} 4 &-3 & 2 \\ 1&2&1 \\ 3 & 1& -2 \end {bmatrix}$ . Find determinant.

Solution : $A=4\begin {vmatrix} 2 & 1\\ 1 &-2 \end {vmatrix}$ – $(-3)\begin {vmatrix} 1 & 1\\ 3 &-2 \end {vmatrix}$ + $2\begin {vmatrix} 1 & 2\\ 3 &1 \end {vmatrix}$ = 4 (-4-1) + 3 (-2-3) + 2 (1-6) = -20-15-10 = -45

Example 4: Let $B=\begin {bmatrix} a &b & c \\ c&a&b \\ b & c& a \end {bmatrix}$

Solution: $A=a\begin {vmatrix} a & b\\ c &a \end {vmatrix}$ – $b\begin {vmatrix} c& b\\ a &a \end {vmatrix}$ + $c\begin {vmatrix} c& a\\ b &c \end {vmatrix}$ = $a(a^{2}-bc)$ – $b(ac-b^{2})$ + $c(c^{2}-ab)$ = $a^{3}+b^{3}+c^{3}-3ab$

Properties of Determinants

The value of determinant remains unchanged if its rows and columns are interchanged.
If any 2 rows or columns are being interchanged the there be the change in sign of determinants.
If any 2 rows or columns are being identical then the value of determinant would be zero.
If the element of a row or column is being multiplied by a scalar then the value of determinant also become a multiple of that constant.
Two determinant can be added if they have 2 identical rows or columns.

Exercise

Find determinant of matrix $D=\begin {bmatrix} 28 &45 &63 \\ 20&34&48 \\ 21 & 36&51 \end {bmatrix}$
Find determinant of matrix $A=\begin {bmatrix} 8&-6 \\ 7&3 \end {bmatrix}$
Find the value of x if $\begin {bmatrix} 5 &5 &x \\ x&5&5 \\5 & 5&4 \end {bmatrix}$ = 0
Evaluate $\begin {bmatrix} a&h &g \\ h&b&f \\g & f&c \end {bmatrix}$
Prove that $\begin {bmatrix} a&a &a \\ a&b&b \\a & b&c \end {bmatrix}$ = a(b-c) (a-b)

Uses of Determinant

Determinant of a 2*2 matrix

Determinant of a 3*3 matrix

Properties of Determinants

Exercise

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