Inverse of Matrix

About Inverse of Matrix

A^-1 is the inverse of Matrix for a matrix A. A simple formula can be used to calculate the inverse of a 2 2 matrix. In addition, we must know the determinant and adjoint of a 3 3 matrix in order to compute its inverse. The inverse of a matrix is another matrix that produces the multiplicative identity when multiplied with the supplied matrix.
The matrix inversion method uses the inverse of a matrix to find the solution to linear equations.

What do you mean by Inverse of Matrix?

The inverse of a matrix is another matrix that yields the multiplicative identity when multiplied with the supplied matrix. A.A^-1 = A^-1•A = I, where I is the identity matrix, is the inverse of a matrix A. An invertible matrix is one whose determinant is non-zero and for which the inverse matrix may be determined.

• Example, the inverse of A = $\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]$ is 112012
• A.A^-1 = 1-102 × $\left[\begin{array}{cc}1& \frac{1}{2}\\ 0& \frac{1}{2}\end{array}\right]$ = I
• A^-1.A = $\left[\begin{array}{cc}1& \frac{1}{2}\\ 0& \frac{1}{2}\end{array}\right]$ × $\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]$= I

Inverse Matrix Formula

The inverse of any real integer a was the number a^-1, so that a times a^-1 equalled 1. As long as the number was not zero, we understood that the inverse of the number was the reciprocal of the number. The inverse of a square matrix A, indicated by A^-1, is the matrix, hence the identity matrix is the product of A and A^-1. The resulting identity matrix will be the same size as matrix A.

• A^-1 = (1/|A|) × Adj(A)

The inverse of a matrix exists only if the determinant of the matrix is a non-zero number, because |A| is in the denominator of the expression. i.e., |A| ≠ 0.

Math Inverse Matrix Formula

• For a matrix A, the inverse matrix formula is: A^-1 = adj(A)/|A|; |A| ≠ 0
• Here A is a square matrix.
• Note: In order for the inverse of a matrix to exist, the given matrix must be square.
• The matrix's determinant should not be equal to zero.

• Inverse of Matrix-Related Terms
• Minor: Every matrix element has a minor that is defined. The determinant produced after deleting the row and column containing this element is the minor of that element.
• Matrix A = $\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}$ the minor of the element a₁₁is: Minor of a₁₁ = $\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|$
• Cofactor: The minor of the exponent of the sum of the row and column elements in order representation of that element is multiplied by -1 to get the cofactor of that element.
• Cofactor of a_ija_ij= (-1)^{i + j} × minor of a_ij.
• Determinant: A matrix's determinant is the matrix's sole unique value representation. The determinant of the matrix can be determined with reference to any row or column of the provided matrix. The sum of the product of the elements and their cofactors in a given row or column of the matrix is the determinant of the matrix.
• Singular Matrix: A singular matrix is defined as a matrix with a determinant value of zero. |A| = 0 for a singular matrix A. There is no such thing as a singular matrix's inverse.
• Non-Singular Matrix: A non-singular matrix is one in which the determinant value is not equal to zero. |A| ≠ 0 for a non-singular matrix Because its inverse can be found, a non-singular matrix is called an invertible matrix.
• Adjoint of Matrix: The adjoint of a matrix is the transpose of the matrix's cofactor element matrix.

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