The Relationship Between a Square Matrix’s Inverse and Its Determinant

a square matrix has an inverse iff

A square matrix has an inverse if and only if its determinant is non-zero

A square matrix has an inverse if and only if its determinant is non-zero.

To understand this, we first need to define what an inverse of a matrix is. Given a square matrix A, its inverse, denoted as A^(-1), is a matrix such that when multiplied with A, the result is the identity matrix, denoted as I. In other words, A * A^(-1) = I.

Now, the determinant of a square matrix is a scalar value that can be computed using various methods. It provides important information about the matrix, specifically regarding its invertibility. A matrix with a determinant of zero is called a singular matrix, which means it does not have an inverse.

To prove that a square matrix has an inverse if and only if its determinant is non-zero, we need to establish both implications:

1. If a square matrix has an inverse, then its determinant is non-zero:
If A has an inverse A^(-1), we can multiply A by A^(-1) to get the identity matrix: A * A^(-1) = I. Taking the determinant of both sides of this equation, we have det(A * A^(-1)) = det(I). Using the property that the determinant of a product is the product of determinants, we get det(A) * det(A^(-1)) = det(I). Since the determinant of the identity matrix is 1, we have det(A) * det(A^(-1)) = 1. From this equation, we can see that det(A) must be non-zero for the equation to hold. Hence, if a matrix has an inverse, its determinant is non-zero.

2. If a square matrix’s determinant is non-zero, then it has an inverse:
To prove this, we can use the concept of the adjugate matrix. The adjugate matrix of a square matrix A, denoted as adj(A), is created by taking the transpose of the cofactor matrix of A. The cofactor of an element a_ij in A is given by (-1)^(i+j) times the determinant of the matrix obtained by removing the i-th row and j-th column of A.

Now, if the determinant of A is non-zero, we can express the inverse of A as A^(-1) = (1 / det(A)) * adj(A). Since det(A) is non-zero, dividing by det(A) is well-defined, and thus A^(-1) exists.

In conclusion, a square matrix has an inverse if and only if its determinant is non-zero.

More Answers:
Understanding the Pythagorean Trig Identity | Deriving the Relationship between Tangent and √2
Understanding the Pythagorean Trig Identity | Exploring the Relationship between Sine and Cosine Functions
The Pythagorean Trig Identity and Its Relationship with Cotangent and Cosecant

Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded

Share:

Recent Posts

Mathematics in Cancer Treatment

How Mathematics is Transforming Cancer Treatment Mathematics plays an increasingly vital role in the fight against cancer mesothelioma. From optimizing drug delivery systems to personalizing

Read More »