Matrix Formulas

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:

C_ij = A_ij + B_ij

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.

C_ij = A_ij – B_ij

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.

C_ij = (A_ik · B_ik)

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Scalar Multiplication

Multiply each element in the matrix by a scalar (a single number).

D_ij = k · A_ij

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] A_ik · B_kj

[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

(A^T)_ij = A_ji

[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]

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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]

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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(A_ij) = (-1)^(i+j) × Minor of A(M_ij)

[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)

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Adjoint of a Matrix Formula

The adjoint of a matrix is found by taking the transpose of the matrix of cofactors.

Adj(A)_ij = C_ij

[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, Cofactor_ij represents the cofactor of the element at the i^th row and j^th 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.

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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]

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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:

A^T· 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,

• Determinant of 3×3 Matrix
• Inverse of 3×3 Matrix

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.