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## Matrix Inverse Explained

The inverse of a matrix is a matrix that “undoes” the original matrix. It is a mathematical operation that finds the matrix that, when multiplied by the original matrix, produces the identity matrix.

The inverse of a matrix is usually denoted by the symbol ” inverse” followed by the original matrix’s name. For example, the inverse of the matrix A is written A inverse.

To find the inverse of a matrix, you must first determine its determinant. The determinant of a matrix is a number that is calculated by taking the determinant of each of the individual rows and columns of the matrix. Once you have the determinant, you can use a special formula to find the inverse of the matrix.

## Definition of Matrix and the Inverse of a Matrix

Matrix: A matrix is a rectangular array of numbers, symbols, or other elements. The numbers in a matrix are usually denoted by lowercase letters, while the symbols are usually denoted by uppercase letters. The number of rows in a matrix is called the matrix’s “height,” while the number of columns is called the matrix’s “width.”

Inverse of a Matrix: The inverse of a matrix is a matrix that “undoes” the original matrix. That is, if the original matrix multiplies a vector by a number, the inverse matrix will multiply the vector by the reciprocal of that number. The inverse matrix is usually denoted by the symbol “A-1.”

## Inverse Matrix Properties

An inverse matrix is a matrix that “undoes” another matrix. That is, if A is an inverse matrix of B, then the product of A and B will equal the identity matrix.

There are a few properties of inverse matrices that are worth noting.

First, the inverse of a matrix is unique. That is, if A is an inverse matrix of B, and B is an inverse matrix of C, then A is also the inverse matrix of C.

Second, the inverse of a matrix is determined by its determinant. That is, if A is an inverse matrix of B, then the determinant of A must be equal to the determinant of B.

Finally, if A is an inverse matrix of B, then the inverse of A will also be an inverse matrix of B.

## There Are Basically 3 Other Properties Of The Inverse As Below:-

**The Inverse Is A One-To-One Function:**

This means that for every input there is only one output. For example, if you input the number 5 into the inverse function, the output would be 1.

**The Inverse Is Commutative:**

This means that the order of the inputs does not affect the output. For example, if you input the number 5 into the inverse function first and then input the number 1, the output would be the same as if you input the number 1 into the inverse function first and then input the number 5.

**The Inverse Is Associative:**

This means that the order of the inputs does not affect the output when there are more than two inputs. For example, if you input the number 5 into the inverse function first, then input the number 2, and then input the number 3, the output would be the same as if you input the number 5 into the inverse function first, then input the number 3, and then input the number 2.

## Need of an Inverse

There is a need for an inverse in mathematics because it allows us to solve problems. For example, if I want to find out what x is if it equals 2 when y is 4, I can use the inverse to solve for x. This is because the inverse allows us to undo operations.

## Calculate the Inverse of a 2×2 Matrix Operations

Inverse of a matrix is the matrix that is its reciprocal matrix. It is also possible to find the inverse matrix algebraically.