# Matrix Formula

A matrix is an ordered arrangement of numbers, expressions, and even symbols, in rows and columns. If the two matrices are of the same size (with respect to their rows and columns), then they can be added, subtracted, and multiplied element by element. Let us learn the matrix formulas along with a few solved examples.

## What Is Matrix Formula?

A matrix is an array of numbers divided into rows and columns, represented in square braces. If you see a 2×2 matrix, then that means the matrix has 2 rows and 2 columns. The matrix formulas are used to calculate the coefficient of variation, adjoint of a matrix, determinant of a matrix, and inverse of a matrix. The matrix formula is useful particularly in those cases where we need to compare results from two different surveys having different values.

### Matrix Formulas

**Formula 1:** Coefficient of variation formula can be given as,

\(M=\begin{bmatrix} m_{11} & m_{12}\\ m_{21} & m_{22} \end{bmatrix}\)

**Formula 2: **The adjoint of a 2×2 matrix formula is given as,

\( adj(M)=\begin{bmatrix} m_{22} & -m_{12}\\ -m_{21} & m_{11} \end{bmatrix}\)

**Formula 3: **The inverse of a 2×2 matrix formula is given as,

\( M^{-1}=\frac{1}{|M|}\times adj(M)\)

**Formula 4: **The determinant is given by the formula:

|M| = \(m_{11}m_{22} – m_{12}m_{21}\)

**Formula 5:** Transpose of a matrix formula

A= \(\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}\)

\(A^T\) =A'= \(\begin{bmatrix}a&d\\b&e\\c&f\end{bmatrix}\)

**Formula 6:** Matrix formula for addition:

A= \(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\)

B= \(\begin{bmatrix}j&k&l\\m&n&o\\p&q&r\end{bmatrix}\)

A+B= \(\begin{bmatrix}a+j&b+k&c+l\\d+m&e+n&f+o\\g+p&h+q&i+r\end{bmatrix}\)

**Formula 7:** Matrix formula for subtraction:

A= \(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\)

B= \(\begin{bmatrix}j&k&l\\m&n&o\\p&q&r\end{bmatrix}\)

A-B= \(\begin{bmatrix}a-j&b-k&c-l\\d-m&e-n&f-o\\g-p&h-q&i-r\end{bmatrix}\)

**Formula 8:** Matrix formula for Multiplication:

A= \(\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}\)

B= \(\begin{bmatrix}g&h\\i&j\\k&l\end{bmatrix}\)

AB= \(\begin{bmatrix}ag+bi+ck&ah+bj+cl\\dg+ei+fk&dh+ej+fl\end{bmatrix}\)

**Formula 9: **For an orthogonal matrix, the product of a matrix and its transpose gives an identity matrix. M × M^{T }= I

M × M^{T }= \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) × \(\begin{bmatrix}a&c\\b&d\end{bmatrix}\) = \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)

## Applications of Matrix Formula

Matrix formulas are commonly used to find solutions for linear equations and calculus, optics, quantum mechanics, and other mathematical functions.

Let us see how to use the matrix formula in the following solved examples section.

## Examples Using Matrix Formula

**Example 1:** Using the matrix formula, determine the determinant of the given matrix.

\(\begin{bmatrix} 3 & 4 \\ 4 & 8 \end{bmatrix}\)

**Solution: **

To Find: The determinant of the matrix.

\(m_{11}\) = 3, \(m_{12}\) = 4, \(m_{21}\) = 4 and,_{ }\(m_{22}\) = 8(given)_{ }

Using matrix formula for determinant,

|M| = \(m_{11}\)\(m_{22}\)–\(m_{12}\)\(m_{21}\)

= (3)(8) - (4)(4)

= 24 - 16

= 8

**Answer: The determinant of the given matrix is 8.**

**Example 2 :** Determine the adjoint of the given 2 x 2 matrix.

\(\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\)

**Solution:**

To Find: The adjoint of a given 2 x 2 matrix

Given: \(m_{11}= 5, m_{12} = 6, m_{21} = 7 and, m_{22} = 8\)(given)_{ }

Using matrix formula for the adjoint of a matrix,

\( adj(M)=\begin{bmatrix} m_{22} & -m_{12}\\ -m_{21} & m_{11} \end{bmatrix}\)

Put all the values,

\( adj(M)=\begin{bmatrix} 8 & - 6\\ - 7 & 5 \end{bmatrix}\)

**Answer: The adjoint of a given is \(\begin{bmatrix} 8 & - 6\\ - 7 & 5 \end{bmatrix}\).**

**Example 3: **Use matrix formula to determine the determinant of the matrix:

\(\begin{bmatrix} 2 & 4 \\ 6 & 5 \end{bmatrix}\)

**Solution:**

To Find: The determinant of the matrix.

\(m_{11}\)= 2, \(m_{12}\) = 4, \(m_{21}\) = 6 and,_{ }\(m_{22}\) = 5(given)_{ }

Using matrix formula for determinant,

|M| = \(m_{11}\)\(m_{22}\)–\(m_{12}\)\(m_{21}\)

= (2)(5) - (4)(6)

= 10 - 24

= -14

**Answer: The determinant of the given matrix is -14.**

## FAQs on Matrix Formula

### What Is a Matrix Formula in Algebra?

The matrix formulas are used to calculate the coefficient of variation, adjoint of a matrix, determinant of a matrix, and inverse of a matrix.

- Coefficient of variation formula can be given as, \(M=\begin{bmatrix} m_{11} & m_{12}\\ m_{21} & m_{22} \end{bmatrix}\)
- The adjoint of a 2×2 matrix formula is given as, \( adj(M)=\begin{bmatrix} m_{22} & -m_{12}\\ -m_{21} & m_{11} \end{bmatrix}\)
- The inverse of a 2×2 matrix formula is given as, \( M^{-1}=\frac{1}{|M|}\times adj(M)\)
- The determinant is given as, |M| = \(m_{11}m_{22} – m_{12}m_{21}\)

### What Is the Identity Matrix Formula?

An identity matrix is a square matrix that comprises of all the main diagonal elements as 1's and all the remaining elements as 0's is called an Identity Matrix, also known as Unit Matrix or Elementary Matrix. For any identity matrix, A×I_{n×n} = A, where A is any square matrix of order n×n.

### What Is the Orthogonal Matrix Formula?

For an orthogonal matrix, the product of a matrix and its transpose gives an identity value. If M is a matrix, M^{T }is its transpose. Their product is an identity matrix with 1 as the values in the leading diagonals.

The orthogonal matrix formula is M × M^{T }= I

### What Are the Applications of Matrix Formula?

Matrix formulas have applications in many fields

- finding solutions for linear equations
- in Calculus,
- in Optics,
- in Quantum Mechanics
- in other mathematical functions.

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