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3x3 Matrix Determinant,Adjoint Matrix,Inverse Matrix Calculator is a free online tool that displays the 3x3 Matrix Determinant,Adjoint Matrix,Inverse Matrix. This online x3 Matrix Determinant,Adjoint Matrix,Inverse Matrix Calculator tool performs the calculation faster, and it displays the result in a fraction of seconds.

The procedure to use the 3x3 Matrix Determinant,Adjoint Matrix,Inverse Matrix Calculator is as follows:

Step 1: Enter a values in the input field

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Step 3: Finally, The 3x3 Matrix Determinant,Adjoint Matrix,Inverse Matrix will be displayed in the output field

What Is Determinant of a Matrix?

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism.

The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or |A|.

How to calculate the Determinant of a Matrix?

In the case of a 2 × 2 matrix the determinant can be defined as:

Similarly, for a 3 × 3 matrix A, its determinant is:

Example:

|D| = 6×(−2×7 − 5×8) − 1×(4×7 − 5×2) + 1×(4×8 − (−2×2)) = 6×(−54) − 1×(18) + 1×(36) = −306

Similarly, for a 4 × 4 matrix A, its determinant is:

What Is Adjugate matrix(Adjoint matrix)?

In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix. It is also occasionally known as adjunct matrix, though this nomenclature appears to have decreased in usage. The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

Let A=[a_ij] be a square matrix of order n . The adjoint of a matrix A is the transpose of the cofactor matrix of A . It is denoted by adj A . An adjoint matrix is also called an adjugate matrix.

How to calculate Adjugate matrix(Adjoint matrix)?

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

What is a cofactor?

A cofactor is a number that is obtained by eliminating the row and column of a particular element which is in the form of a square or rectangle. The cofactor is preceded by a negative or positive sign based on the element’s position.

What is a Cofactor matrix?

Cofactor matrix is a matrix having the cofactors as the elements of the matrix.

How to calculate the Cofactor Matrix?

First, find the minor of each element of the matrix by excluding the row and column of that particular element, and then taking the remaining part of the matrix.

Secondly, find the minor element value by taking the determinant of the remaining part of the matrix.

The third step involves finding the co-factor of the element by multiplying the minor of the element with -1 to the power of position values of the element.

The fourth steps involves forming a new matrix with the co-factors of the elements of the given matrix, to form the co-factor matrix.

Cofactor Formula:

Let A be any matrix of order n x n and M_ij be the (n – 1) x (n – 1) matrix obtained by deleting the ith row and jth column. Then, det(M_ij) is called the minor of a_ij. The cofactor C_ij of aij can be found using the formula:

C_ij = (−1)^i+jdet(M_ij)

Thus, cofactor is always represented with +ve (positive) or -ve (negative) signs.

What is a Inverse matrix?

If A is a non-singular square matrix, there is an existence of n x n matrix A-1, which is called the inverse matrix of A such that it satisfies the property:

AA^-1 = A^-1A = I, where I is the Identity matrix

How to calculate the Inverse matrix?

A^-1= adj(A)/det(A) = adj(A)/|A|

where

A is a square matrix.

adj(A) is the adjoint matrix of A.

|A| is the determinant of A.

Solved Examples: Calculate the Cofactor,Cofactor Matrix,Matrix Determinant, Adjoint Matrix, Inverse Matrix of Matrix A

Solution:

The Cofactor,Cofactor Matrix is:

Then Calculate the Adjoint matrix:

Adjoint of a square matrix is the transpose of its cofactor matrix:

Now find the transpose of A_ij .

adj A=(A_ij)^T

Then Calculate the Matrix Determinant:

Please refer to the beginning of this article "How to calculate the Determinant of a Matrix?"

|A| = 2

Then Calculate the Inverse Matrix:

A^-1 = adj(A)/|A| = adj(A)/2

So The Adjoint Matrix,Matrix Determinant,Inverse Matrix of Matrix A is: