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.

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