The Invertibility of the Product of Two Invertible Matrices
The product of two matrices is a fundamental concept in linear algebra, and it plays a crucial role in various applications, including matrix multiplication, eigenvalue decomposition, and linear transformations. However, the question remains whether the product of two invertible matrices is also invertible. In this article, we will explore the properties of the product of two invertible matrices and provide a direct answer to this question.
Matrix Multiplication and Invertibility
Matrix multiplication is a fundamental operation in linear algebra, and it is defined as the product of two matrices A and B, where the element in the ith row and jth column of the resulting matrix C is given by:
C[i, j] = ∑[A[i, k] * B[k, j]]
For a matrix to be invertible, it must have a non-zero determinant. In other words, the determinant of a matrix A is non-zero if and only if the matrix is invertible.
Properties of Invertible Matrices
An invertible matrix has the following properties:
- Non-zero determinant: The determinant of an invertible matrix is non-zero.
- Non-singular: The matrix is non-singular, meaning that it has an inverse.
- Square matrix: The matrix is a square matrix, meaning that it has the same number of rows and columns.
Product of Two Invertible Matrices
Let A and B be two invertible matrices, where A is a 2×2 matrix and B is a 2×2 matrix. We can represent these matrices as:
A = | a11 a12 |
| a21 a22 |
B = | b11 b12 |
| b21 b22 |
The product of A and B is given by:
AB = | a11b11 + a12b21 a11b12 + a12b22 |
| a21b11 + a22b21 a21b12 + a22b22 |
For A and B to be invertible, their determinants must be non-zero. Let’s calculate the determinants of A and B:
det(A) = a11b11 + a12b21
det(B) = b11b22 – b12b21
Since A and B are invertible, their determinants must be non-zero. Therefore, we have:
a11b11 + a12b21 ≠ 0
b11b22 – b12b21 ≠ 0
Product of Two Invertible Matrices is Invertible
From the above calculations, we can see that the product of two invertible matrices A and B is also invertible. This is because the determinants of A and B are non-zero, which means that the matrix is non-singular.
Example
Let’s consider two 2×2 invertible matrices:
A = | 1 2 |
| 3 4 |
B = | 5 6 |
| 7 8 |
The product of A and B is given by:
AB = | 15 + 27 16 + 28 |
| 35 + 47 36 + 48 |
AB = | 27 22 |
| 43 50 |
The determinant of AB is non-zero, which means that the matrix is invertible.
Conclusion
In conclusion, the product of two invertible matrices is also invertible. This is because the determinants of the matrices are non-zero, which means that the matrix is non-singular. The product of two invertible matrices is a fundamental concept in linear algebra, and it plays a crucial role in various applications, including matrix multiplication, eigenvalue decomposition, and linear transformations.
Table
| A | B | |
|---|---|---|
| det(A) | a11b11 + a12b21 | b11b22 – b12b21 |
| det(B) | b11b22 – b12b21 | a11b22 – a12b21 |
Note: The table is not exhaustive, but it illustrates the relationship between the determinants of A and B and the product AB.