Misc Examples Matrices and Determinants

Various arithmetic operations involving an array of elements are carried out using matrices and determinants. Matrices can be solved using arithmetic operations such as addition, subtraction, multiplication, and finding the inverse. In this article, we'll use the ideas of matrices and determinants, their properties, the determinant formula, and the distinction between a matrix and a determinant to solve various miscellaneous cases. 

Additionally, there are many fields in which matrices and determinants are used. Simply put, a matrix is a means to arrange integers into rows and columns. A determinant is a real number associated with each square matrix.

What are Matrices and Determinants?

Matrices: A matrix is a collection of components shown as rows and columns. Determinants are regarded as scalar matrix components. The number of rows and columns in a matrix serves as a representation of the matrix's order. 

Let A be a matrix then

$A = \begin{bmatrix} a\end{bmatrix}_{m \times n} = \begin{bmatrix} a_{11} &a_{12} . & . & & . & a_{1n} \\a_{21} & a_{22} & . & . & . & a_{2n} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\a_{m1} & a_{m2} & . & . & . & a_{mn} \\ \end{bmatrix}$

Where $a_{ij}=$  Element of the matrix.

m = numbers of rows

n = number of columns

And $m \times n =$  order of matrix

Determinants:  With each $n \times n$ square matrix, we associate a real number called its determinant.

If A is a matrix, we denote its determinant by $ \begin{vmatrix}A \end{vmatrix}$

Let $A = \begin{bmatrix} 2 & -1& 3 \\3 & 1 & 2 \\-1 & 2 & -3 \\ \end{bmatrix}$

Then $ \begin{vmatrix}A\end{vmatrix}= \begin{vmatrix}2 & -1 & 3 \\3 & 1 & 2 \\1 & 2 & -3 \\ \end{vmatrix}$  which is defined only for a square matrix.

Types of Determinants

Based on the types of square matrix, determinants are of 3 kinds

1. First order determinant 

$ \begin{bmatrix} A\end{bmatrix}=\begin{vmatrix} A\end{vmatrix}$

2. Second order determinant - For $2 \times 2$ matrix  

$\begin{bmatrix} 2 & 3 \\ 5 & -4 \\ \end{bmatrix}=\begin{vmatrix} 2& 3 \\ 5 & -4 \\ \end{vmatrix}$

3. Third order determinant - For $3 \times 3$ matrix 

$\begin{bmatrix} 1& -2 & 3 \\2 1 & -1 & \\-2 & -1 & 2 \\ \end{bmatrix}=\begin{vmatrix} 1 & -2 & 3 \\ 2 & 1 &-1 \\ -2 &-1 & 2 \\ \end{vmatrix}$

Determinant Formula

It is a mathematical expression that is defined only for square matrices.

It is used to find the determinant of a matrix.

The determinant formula for $\begin{bmatrix} a & b \\c &d \\ \end{bmatrix}_{2 \times 2}$  given by 

$D_{2\times 2}= ad - bc$

Determinant formula for $\begin{bmatrix}a & b & c \\d & e & f \\g & h & i \end{bmatrix}_{3 \times 3}$ is given by

$D_{3 \times 3}= a\left ( ei-fh \right )-b\left ( di-fg \right )+c\left ( dh-eg \right )$

Properties of Matrices and Determinants

Properties of Matrices

1. Additive properties: In addition to matrices, we add the corresponding elements. If A is a matrix of order$2 \times 2$, we can only add it to another by the same order i.e. $2 \times 2$ matrix i.e. the number of rows and columns must be the same when adding matrices.

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -4 \\ \end{bmatrix}$

And  B =  $\begin{bmatrix} 7 & 4 \\-3 & 5 \end {bmatrix}$

Then  $A + B =  \begin{bmatrix} 2 + 7 & 3 + 4 \\5 + \left ( -3 \right )  & \left ( -4 + 5 \right ) \\ \end{bmatrix}$

$= \begin{bmatrix} 9 & 7 \\ 2 & 1 \end{bmatrix}$

 If A, B and C are matrices then the addition of matrices will follow the following properties

 $A + B = B + A$

 $(A + B) + C = A + (B + C)$

 $K (A + B) = K A + K B$

2. Multiplicative properties: We define the multiplication of two matrices as below:

Let A be an m x n matrix and B an $n \times p$ matrix.

The product AB is that m x p matrix C with element $C_{ij}$  given by

$C_{ij} = \sum_{k = 1}^{n}a_{ik} \hspace{.1cm}b_{kj}\hspace{1cm} i = 1,2,...m,j = 1,2...p$  

Product of two matrices are defined only when the number of columns in the first matrix is equal to the number of rows in the second matrix. For example,

If $A =  \begin{bmatrix} 2 &-1 & 3 \\1 & -2 & -1 \\ \end{bmatrix}$

And $B = \begin{bmatrix} 3 & -1 \\1 & 2 \\-1 & 1 \\ \end{bmatrix}$

Then AB = $\begin{bmatrix}2 & -1 \\2 & -6 \end{bmatrix}$

If A, B and C are matrices then the operation of multiplication will follow the following properties.

$ AB \neq BA $

$\left ( AB\right )C = A\left ( BC \right ) $

$ A \left ( B + C \right ) = AB + BC$

$\left ( A + B \right )C = AC + BC$

For a square matrix A

$AI=IA=A$

3. Transpose properties: We define the transpose of a matrix as below:

If A is an $m \times n$ matrix, the transpose of A is that $n \times m$ matrix obtained from A by interchanging its rows and columns; we use the symbol $A^{T}$ for the transpose of A. For example- If $A =\begin{bmatrix} 2 & -1 & 3 \\ 1 & -2 & -1 \\ \end{bmatrix}$ then $A^{T} =         \begin{bmatrix} 2 & 1 \\ -1 & -2 \\ 3 & -1 \\ \end{bmatrix}$

For matrices A and B transpose will follow the following properties.

$ \left ( A^{T} \right )^{T} = A$

$ \left ( kA\right )^{T}=kA^{T}$

$ \left ( A + B \right )^{T}=A^{T} + B^{T}$ 

$ \left ( AB \right )^{T} = B^{T}A^{T}\left ( A^{T} \right )^{T}=A $

If $A^{T}=A$, then A is said to be a symmetric matrix.

If $A^{T}=-A$, then A is said to be a skew-symmetric matrix.

4. Inverse properties: We find the inverse of the matrix by the following method 

Let $A= \begin{bmatrix} 4 & 7 \\ 2 & 6 \\ \end{bmatrix}$

be a square matrix then

$Adj A = Transpose of cofactor matrix$

$=  \begin{bmatrix} 6 & -7 \\ -2& 4 \\ \end{bmatrix}$

$ \begin{vmatrix}A\end{vmatrix}$

$= 4\times 6-14$

$ = 24 -14$

Then

$ A^{-1} = \dfrac{AdjA}{\begin{vmatrix} A\end{vmatrix}}$

$=  \dfrac{1}{10}\begin{bmatrix} 6 & -7 \\ -2 & 4 \\ \end{bmatrix}$

$=   \begin{vmatrix} 0.6 &-0.7 \\ -0.2 &0.4 \\ \end{vmatrix}$

For a matrix A inverse follows the following properties

$ A^{-1}$is unique i.e. there is only one inverse of a matrix

$\left ( A^{-1}\right )^{-1} = A$

$\left ( kA \right )^{-1} = \dfrac{1}{k}A^{-1}$

$\left ( A^{-1} \right )^{-T} = \left ( A^{T} \right )^{-1}$

$\left ( A + B \right )^{-1}= A^{-1}+ B^{-1}$

$\left ( AB \right )^{-1}= B^{-1}A^{-1}$

Properties of Determinants

1. Interchanging rows with columns

$\begin{bmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} &c_{2} & c_{3} \\ \end{bmatrix}= \begin{vmatrix} a_{1} & b_{1} &c_{1} \\ a_{2} &b_{2} & c_{2} \\ a_{3} &b_{3} &c_{3} \\ \end{vmatrix}$

2. Interchanging any two rows/columns

$\begin{bmatrix} a_{1} &a_{2} &a_{3} \\ b_{1} &b_{2} &b_{3} \\ c_{1} &c_{2} & c_{3} \\ \end{bmatrix}= -\begin{vmatrix} b_{1} & b_{2} &b_{3} \\ a_{1} & a_{2} &a_{3} \\ c_{1} &c_{2} & c_{3} \\ \end{vmatrix}$

3. When any two rows/columns are equal

$\begin{bmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} &b_{2} & b_{3} \\ b_{1} &b_{2} &b_{3} \\ \end{bmatrix}= 0$

4.  If A is $n \times n$ matrix, then A will follow the following properties.

$\begin{vmatrix}A^{T}\end{vmatrix}= \begin{vmatrix} A\end{vmatrix}$

 $\begin{vmatrix} A^{-1}\end{vmatrix}= \dfrac{1}{A}$

$\begin{vmatrix} A^{-1}\end{vmatrix}= \dfrac{1}{A}$

$\begin{vmatrix} kA\end{vmatrix}^{n}= k^{n}\begin{vmatrix} A\end{vmatrix}$

Where n = order of matrix

Similarly,

$\begin{vmatrix} -A\end{vmatrix}= \begin{vmatrix} \left ( -1 \right )\times A\end{vmatrix}= \left ( -1 \right )^{n}\times \begin{vmatrix} A\end{vmatrix}$

$\begin{vmatrix} AdjA\end{vmatrix}= A^{n - 1}$

Applications of Matrices and Determinants

• Both are used to calculate the inverse of a matrix to solve a system of linear equations.

• Both are used for solving business problems.

• Both are used in finding the volume of a parallelepiped, the area of a triangle, the area of the parallelogram, and engineering.

Solved Examples

Example 1: Find the determinant of a given symmetric matrix.

$A = \begin{bmatrix} 1 & a_{1} & a_{2} \\ a_{1}& 1 & a_{3} \\ a_{2} &a_{3} & 1 \\ \end{bmatrix}_{3 \times 3}$

Solution: 

$\begin{vmatrix} A\end{vmatrix}= \left (a_{1}- a_{2} \right )-a_{1}\left ( a_{1}-a_{3} \right )+a_{2}\left ( a_{1}a_{2}-a_{1}a_{3} \right )$

$= a_{1}-a_{2}-a_{1}^{2}+a_{1}a_{3}+a_{1}a_{2}a_{3}-a_{1}a_{2}a_{3}$

$= a_{1}\left ( 1-a_{1}+a_{3} \right )-a_{2}$

Example 2: For what value of x the matrix $\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 &-4 \\ 3& 4 & x + 5 \\ \end{bmatrix}$ is  skew-symmetric matrix.

Solution: Since the given matrix is skew-symmetric therefore,

$\begin{bmatrix}0 & 2 &-3 \\-2 & 0 &-4 \\3 & 4 &x+5 \\ \end{bmatrix}$

$=-\begin{bmatrix} 0 &-2 & 3 \\ 2& 0 & 4 \\ -3 & -4 &x+5 \\ \end{bmatrix}$ 

Therefore $x+5=-x-5$

$ \Rightarrow x=-5$

Hence the values of x is $-5$

Interesting Facts

• As a formula$ \begin{vmatrix} A\end{vmatrix}$, the parallel indicates “ determinant of A”.

• Determinant can be zero.

Conclusion

The article summarises the matrices and determinants, properties of matrices and determinants, determinant formula, difference between matrix and determinant, application of matrices and determinants and examples.

We know that  A matrix is a way for numbers to organise into rows and columns, whereas a determinant is a real number associated with each square matrix. While solving miscellaneous examples, we require to find the inverse of a matrix. The inverse of a matrix necessitates finding the cofactors of each element of either row or column of the matrix. 

Practice Questions

1. $A = \begin{bmatrix} 0 & c & -b\\ -c & 0 & a \\ b & -a & 0 \\ \end{bmatrix} then \left (a^{2}+b^{2}-c^{2} \right )$     value of $ \begin{vmatrix}A \end{vmatrix}$?

1. $abc$

2. $0$

Answers:

$ \begin{vmatrix}A \end{vmatrix} = 0\times \left ( a^{2} \right )-c\left ( -ab \right )-b\left ( ac \right )$

$= 0 + abc - abc$

$= 0$

B. $0$

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