Matrix Formulas are mathematical expressions that involve operations on matrices. A matrix is a collection of numbers arranged in rows and columns. The formulas of a matrix include adding, subtracting, multiplying, or finding determinants. They’re used in various fields, including math, physics, and computer science.
In this article, we will understand the various formulas of the Matrix with examples.
What is a Matrix?
A matrix is a mathematical structure of numbers, expressions, or symbols arranged in rectangular or square and organized in rows and columns. Each number in the matrix has a specific position, determined by its row and column. The matrices are denoted in capital letters, and their elements are represented in lowercase letters with subscripts showing their position in the matrix. For example, in a matrix A with elements aij, where i represents the row and j represents the column:
[Tex]\begin{bmatrix}
a_{11} & a_{21} & \dots & a_{m1} \\
a_{12} & a_{22} & \dots & a_{m2} \\
a_{13} & a_{23} & \dots & a_{m3} \\
\vdots & \vdots & \ddots & \vdots \\
a_{1n} & a_{2n} & \dots & a_{mn} \\
\end{bmatrix}
[/Tex]
In this example, the matrix A has m rows and n columns, making it an m×n matrix. If the number of rows and columns is the same (m=n), it’s a square matrix. Otherwise, it’s called a rectangular matrix.
A matrix formula is a way to express and perform mathematical operations using matrices. Matrices are like organized grids of numbers or symbols arranged in rows and columns. The formula involves specific rules for combining these matrices to get a result.
The formula will define how to manipulate these matrices, like adding them together, multiplying them, or applying other operations.
For example, a basic matrix formula can be represented as: Cij = Aij + Bij
Where, A and B are matrices, and C is the result of adding them together. Each element in C is the sum of the corresponding elements in A and B.
Matrix formulas involve various operations and expressions that manipulate matrices. Some common matrix formulas are:
- Matrix Formula For Addition
- Matrix Formula For Subtraction
- Matrix Multiplication Formula
- Formula for Transpose of a Matrix
- Formula for Inverse Matrix
- Formula for Cofactor of Matrix
- Formula for Adjoint of a Matrix
- Formula for Determinant of Matrix
- Orthogonal Matrix Formula
- Identity Matrix Formula
Matrix Formula For Addition
Adding matrices is like combining them. To add matrices, you take matching elements from each matrix and add them together. The formula for matrix addition:
Cij = Aij + Bij
This formula is applied to each element in the matrices to create a new matrix with the same dimensions.
For example, given two Matrices A and B
[Tex]A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}
[/Tex]and [Tex]B = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}
[/Tex]
On adding A and B:
[Tex]C = \begin{bmatrix} 2+1 & 4+2 \\ 1+3 & 3+5 \end{bmatrix}
[/Tex]
∴ [Tex]C = \begin{bmatrix} 3 & 6 \\ 4 & 8 \end{bmatrix}
[/Tex]
Matrix Formula For Subtraction
Subtract the elements of one matrix from the corresponding elements of another. The result is a new matrix with the same size.
Cij = Aij – Bij
For example there two matrices given
[Tex]A = \begin{bmatrix} 4 & 7 \\ 2 & 5 \end{bmatrix}
[/Tex] and [Tex]B = \begin{bmatrix} 1 & 3 \\ 1 & 2 \end{bmatrix}
[/Tex]
On subtracting B from A:
[Tex]C = \begin{bmatrix} 4-1 & 7-3 \\ 2-1 & 5-2 \end{bmatrix}
[/Tex]
∴ [Tex]C = \begin{bmatrix} 3 & 4 \\ 1 & 3 \end{bmatrix}
[/Tex]
Matrix Formula For Multiplication
To get each element in the product matrix, you take the dot product of a row from the first matrix and a column from the second matrix.
Cij = (Aik · Bik)
Learn, Matrix Multiplication
Scalar Multiplication
Multiply each element in the matrix by a scalar (a single number).
Dij = k · Aij
k = 3
[Tex]A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}
[/Tex]
[Tex]D = \begin{bmatrix} 3 \cdot 2 & 3 \cdot 4 \\ 3 \cdot 1 & 3 \cdot 3 \end{bmatrix}
[/Tex]
∴ [Tex]D = \begin{bmatrix} 6 & 12 \\ 3 & 9 \end{bmatrix}
[/Tex]
Matrix Multiplication
To multiply two matrices, each element in the resulting matrix is the sum of the products of corresponding elements from a row in the first matrix and a column in the second matrix.
[Tex]C_{ij} = \sum_{k=1}^{n}
[/Tex] Aik · Bkj
[Tex]A = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}
[/Tex] and [Tex]B = \begin{bmatrix} 5 & 2 \\ 6 & 8 \end{bmatrix}
[/Tex]
Multiplying A and B:
[Tex]C = \begin{bmatrix} (2 \cdot 5 + 3 \cdot 6) & (2 \cdot 2 + 3 \cdot 8) \\ (4 \cdot 5 + 1 \cdot 6) & (4 \cdot 2 + 1 \cdot 8) \end{bmatrix}
[/Tex]
∴ [Tex]C = \begin{bmatrix} 28 & 26 \\ 26 & 20 \end{bmatrix}
[/Tex]
Transpose of a Matrix Formula
The transpose of a matrix is like flipping it over its main diagonal. Swap or interchange the rows and columns of the original matrix to get the transpose
(AT)ij = Aji
[Tex]A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}
[/Tex]
∴ Transpose of A is,
[Tex]A^T = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}
[/Tex]
Learn, Transpose of a Matrix
Inverse Matrix Formula
Finding the inverse of a matrix is like finding its opposite in terms of multiplication. If a matrix A has an inverse, multiplying A by its inverse gives the identity matrix.
For a matrix A and its inverse A⁻¹, the formula is
A · A-1 = A-1 · A = I
[Tex]A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)
[/Tex]
[Tex]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}
[/Tex]
the determinant |A|
|A| = (2 × 4) – (3 × 1) = 5
Find the adjugate matrix [adj(A)]:
[Tex]\text{adj}(A) = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}
[/Tex]
[Tex]A^{-1} = \frac{1}{5} \cdot \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}
[/Tex]
∴ [Tex]A^{-1} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}
[/Tex]
Learn, Inverse Matrix
Cofactor of Matrix Formula
To find the cofactor of a matrix element, multiply the determinant of the matrix without the row and column of that element by -1(i+j), where (i) and (j) are the row and column indices.
Cofactor(Aij) = (-1)(i+j) × Minor of A(Mij)
[Tex]A = \begin{bmatrix} 3 & 1 & 4 \\ 1 & 5 & 9 \\ 2 & 6 & 5 \end{bmatrix}
[/Tex]
To find the cofactor of the element in the second row and third column (A23):
Cofactor(A23) = (-12+3) × det(Matrix without row 2 and column 3)
= [Tex](-1)^5 \times \text{det}\begin{bmatrix} 3 & 1 \\ 2 & 6 \end{bmatrix}
[/Tex]
= -1 × (3 × 6 – 1 × 2)
= -1 × (18 – 2)
= -1 × 16
= -16
∴ Cofactor of (A23) is (-16)
Learn, Cofactor of Matrix
Adjoint of a Matrix Formula
The adjoint of a matrix is found by taking the transpose of the matrix of cofactors.
Adj(A)ij = Cij
[Tex]\text{Adjoint}(A) = \begin{bmatrix} \text{Cofactor}_{11} & \text{Cofactor}_{21} & \ldots & \text{Cofactor}_{n1} \\ \text{Cofactor}_{12} & \text{Cofactor}_{22} & \ldots & \text{Cofactor}_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ \text{Cofactor}_{1n} & \text{Cofactor}_{2n} & \ldots & \text{Cofactor}_{nn} \end{bmatrix}^T
[/Tex]
In this formula, Cofactorij represents the cofactor of the element at the ith row and jth column in the original matrix.
For example in matrix (A)
[Tex]A = \begin{bmatrix} 3 & 1 & 4 \\ 1 & 5 & 9 \\ 2 & 6 & 5 \end{bmatrix}
[/Tex]
Find the adjoint of \(A\):
[Tex]\text{Adjoint}(A) = \begin{bmatrix} \text{Cofactor}_{11} & \text{Cofactor}_{21} & \text{Cofactor}_{31} \\ \text{Cofactor}_{12} & \text{Cofactor}_{22} & \text{Cofactor}_{32} \\ \text{Cofactor}_{13} & \text{Cofactor}_{23} & \text{Cofactor}_{33} \end{bmatrix}^T
[/Tex]
First, calculate the cofactor for each element:
[Tex]text{Cofactor}_{11} = (-1)^{1+1} \times \text{det}(\text{Matrix without row 1 and column 1}) = \text{det}\begin{bmatrix} 5 & 9 \\ 6 & 5 \end{bmatrix} = (5 \times 5 – 9 \times 6) = -21
[/Tex]
Similarly, calculate all the cofactors for each element.
Now, assemble the cofactors into a matrix
[Tex]\text{Matrix of Cofactors} = \begin{bmatrix} -21 & 24 & -5 \\ 3 & -30 & 12 \\ -3 & 18 & -15 \end{bmatrix}
[/Tex]
Finally, Transpose the matrix of cofactors to get the adjoint:
[Tex]\text{Adjoint}(A) = \begin{bmatrix} -21 & 3 & -3 \\ 24 & -30 & 18 \\ -5 & 12 & -15 \end{bmatrix}
[/Tex]
[Tex]\text{Adjoint}(A) = \begin{bmatrix} -21 & 3 & -3 \\ 24 & -30 & 18 \\ -5 & 12 & -15 \end{bmatrix}
[/Tex]
Learn, Adjoint of a Matrix
Determinant of Matrix Formula
The determinant of a matrix is a value associated with it. The formula for the determinant of a matrix A is often recursive, involving cofactors. The determinant is often represented as det(A).
To find the determinant of a 2×2 matrix, multiply the diagonal elements and subtract the product of the off-diagonal elements. For a larger matrix, use expansion by minors or other methods.
For a 2×2 matrix:
[Tex]\text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix}
[/Tex] = (a × d) – (b × c)
In a matrix [Tex]B = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}
[/Tex]
det(B) = (2 × 4) – (3 × 1)
= 8 – 3
= 5
∴Determinant of matrix B is 5.
Learn, Determinant of Matrix
Identity Matrix Formula
The identity matrix is like a special matrix where when you multiply it with another matrix, the result is the same matrix. The formula for an identity matrix I with size n×n is represented as:
[Tex]I_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}
[/Tex]
For example in the Identity Matrix of size 2×2:
[Tex]I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
[/Tex]
Learn, Identity Matrix
Orthogonal Matrix Formula
In an orthogonal matrix the rows and columns are perpendicular (or orthogonal) unit vectors.
The formula for an orthogonal matrix is related to its transpose:
AT· A = I
For a 2×2 orthogonal matrix:
[Tex]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
[/Tex]
For this matrix to be orthogonal:
[Tex]A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}
[/Tex] and, [Tex]A^{-1} = \frac{1}{ad – bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
[/Tex]
So, for the matrix to be orthogonal:
[Tex]\begin{bmatrix} a & c \\ b & d \end{bmatrix} = \frac{1}{ad – bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
[/Tex]
This condition should be satisfied for every element in the matrix. If it does, then the matrix is orthogonal.
Learn, Orthogonal Matrix
Also Read,
Various applications of Matrix formulas are added below as,
- Solving Systems of Equations: Matrices are used to efficiently solve problems involving multiple equations and variables.
- Computer Graphics: Matrices are applied to perform transformations such as scaling and rotation in computer graphics.
- Linear Regression in Statistics: Matrices are utilized to analyze relationships between variables and make predictions based on collected data.
- Engineering Applications: Matrices are employed in structural analysis to model and solve complex systems.
- Physics (Quantum Mechanics): Matrices play a crucial role in representing and manipulating quantum states.
- Cryptography: Matrices are used in encryption and decryption algorithms for secure communication.
Example 1: In the given Matrices [Tex]A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}
[/Tex] and [Tex]B = \begin{bmatrix} 4 & 1 \\ 2 & 6 \end{bmatrix}
[/Tex]
- Find the result of adding matrix A to matrix B using the matrix addition formula.
- Determine the result of subtracting matrix B from matrix A using the matrix subtraction formula.
Solution:
Matrix Addition
Cij = Aij + Bij
[Tex]C = \begin{bmatrix} 2+4 & 5+1 \\ 1+2 & 3+6 \end{bmatrix}
[/Tex]
[Tex]C = \begin{bmatrix} 6 & 6 \\ 3 & 9 \end{bmatrix}
[/Tex]
Matrix Subtraction
Dij = Aij – Bij
[Tex]D = \begin{bmatrix} 2-4 & 5-1 \\ 1-2 & 3-6 \end{bmatrix}
[/Tex]
[Tex]D = \begin{bmatrix} -2 & 4 \\ -1 & -3 \end{bmatrix}
[/Tex]
Result of adding A to B is:
[Tex]C = \begin{bmatrix} 6 & 6 \\ 3 & 9 \end{bmatrix}
[/Tex]
And the result of subtracting B from A is:
[Tex]D = \begin{bmatrix} -2 & 4 \\ -1 & -3 \end{bmatrix}
[/Tex]
Example 2: Consider the matrix [Tex]A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}
[/Tex]. Find the transpose of matrix ( A ).
Solution:
Given,
[Tex]A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}
[/Tex]
To find the transpose of matrix ( A ), we swap its rows and columns
[Tex] A^T = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}
[/Tex]
Transpose of matrix ( A ) is [Tex]\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}
[/Tex]
Q1. Given matrices [Tex]A = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}
[/Tex] and [Tex]B = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}
[/Tex], find the sum matrix C.
Q2. Subtract matrix [Tex]B = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}
[/Tex] from matrix [Tex]A = \begin{bmatrix} 6 & 8 \\ 3 & 5 \end{bmatrix}
[/Tex] to get matrix C.
Q3. Multiply matrix [Tex]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}
[/Tex] by scalar k to get matrix B, where k = 2
Q4. Find the transpose of matrix [Tex]A = \begin{bmatrix} 5 & 7 & 2 \\ 1 & 8 & 4 \end{bmatrix}
[/Tex]
Q5. Determine if the following matrix [Tex]A = \begin{bmatrix} 0.8 & -0.6 \\ 0.6 & 0.8 \end{bmatrix}
[/Tex] is orthogonal.
1. What are Matrix Formulas?
A matrix formula is a way of expressing mathematical relationships using matrices. It involves arranging numbers in rows and columns to represent data or perform calculations.
2. What is a Matrix?
A matrix is a structured arrangement of numbers in rows and columns. It is a mathematical tool commonly used in various fields to organize and manipulate data.
3. How are Matrix Formulas Applied in Real-Life Situations?
Matrix formulas find practical use in real-life scenarios such as computer graphics, data analysis, and solving systems of equations. They provide a structured approach to handling and analyzing complex data sets.
4. How to Add Two Matrices?
To add two matrices we add each corresponding elements of the matrix with its respective elements.
5. What is the Identity Matrix Formula?
Identity matrix formula is, for any identity matrix [I], [A]×[I] = A
6. What Is the Orthogonal Matrix Formula?
Orthogonal matrix formula is, for any orthogonal matrix [M], [M]×[M]T = I
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