Invertible Matrix

1What are Invertible Matrices

Invertible matrices are defined as matrices whose inverse exists. They are a non-singular, square matrix of order n×n. If A is an invertible matrix and its inverse is  B, then 

A×B = I_n

where I_n is the inverse matrix.

What are Properties of an Invertible Matrix?

Some of the important properties of invertible matrices are,

• The inverse of any matrix is unique.
• x = A^-1B is the solution of the equation, Ax = B
• Det (A^-1) = (Det A)^-1
• (cA)^-1 = 1/c.A^-1

What is the condition for an Invertible Matrix?

For any square matrix A, the condition for the matrix to be invertible is, there exists a unique square matrix B such that,

A×B = I_n

What is a Non-Invertible Matrix?

A matrix whose inverse does not exists are called a non-invertible matrix. All the singular matrices and the matrix which are not square are non-invertible matrices.

What is Inverse Matrix Theorem?

The inverse matrix theorems are discussed below,

• Every invertible matrix has a unique inverse.
• For two matrices A and B of the same order and if their multiplication AB exists. Then (AB)^-1= B^-1A^-1.

Can a Matrix be Invertible if its determinant is 0?

No, a matrix can not be invertible if its determinant is 0. As we know that a matrix is invertible if the inverse of the matrix exists and the inverse of matrix A is calculated as,

A^-1 = adj A/|A|

And if the determinant of the matrix is zero then its inverse does not exist and hence it is invertible.

What is Invertibility of a Matrix?

A square matrix is invertible iff its determinant is zero. Let us suppose we take a square matrix A of order n×n, then the inverse of the matrix exist only when determinant of matrix A is zero, i.e. |A| = 0.

How do you Determine the Invertibility of a Matrix?

For any matrix if its determinant is zero then the matrix is not invertible or else the matrix is invertible.

What do Eigenvalues say about Invertibility?

For an invertible matrix we cannot have an eigenvalue equal to zero.

Is A 2×2 matrix always invertible?

No, a 2×2 matrix is not always invertible. It is only invertible if the determinant of the 2×2 matrix is zero.

Invertible matrices are defined as the matrix whose inverse exists. We define a matrix as the arrangement of data in rows and columns, if any matrix has m rows and n columns then the order of the matrix is m × n where m and n represent the number of rows and columns respectively. 

We define invertible matrices as square matrices whose inverse exists. They are non-singular matrices as their determinant exists. There are various methods to calculate the inverse of the matrix.

In this article, we will learn about, What are Invertible Matrices? Invertible Matrices Examples, Invertible Matrix Theorems, Invertible Matrix Determinant, and others in detail.