If A and B are two invertible square matrices of same order, then what is (AB) -1 equal to?
B -1 A-1
Understanding Matrix Inverses and Products
When working with matrices, the inverse of a matrix is a fundamental concept. An invertible matrix (or non-singular matrix) is a square matrix A for which there exists a matrix denoted as \(A^{-1}\), such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix of the same order.
The question asks for the inverse of the product of two invertible square matrices, A and B, of the same order. We need to find what \((AB)^{-1}\) is equal to.
The Inverse of a Matrix Product \((AB)^{-1}\)
There is a specific property that governs the inverse of a product of matrices. If A and B are two invertible matrices of the same order, the inverse of their product AB is given by the product of their inverses in reverse order.
The property states:
\[ (AB)^{-1} = B^{-1}A^{-1} \]
This means that to find the inverse of AB, you first find the inverse of B (\(B^{-1}\)), then find the inverse of A (\(A^{-1}\)), and finally multiply \(B^{-1}\) by \(A^{-1}\) in that specific order.
Deriving the Property: \((AB)^{-1} = B^{-1}A^{-1}\)
We can verify this property by multiplying \((AB)\) by \((B^{-1}A^{-1})\) and checking if the result is the identity matrix \(I\). Using the associativity of matrix multiplication:
\[ (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} \]
Since B is invertible, we know that \(BB^{-1} = I\), the identity matrix. Substituting this into the equation:
\[ A(BB^{-1})A^{-1} = A(I)A^{-1} \]
Multiplying by the identity matrix does not change the matrix:
\[ A(I)A^{-1} = AA^{-1} \]
Since A is invertible, we know that \(AA^{-1} = I\), the identity matrix:
\[ AA^{-1} = I \]
Thus, we have shown that \((AB)(B^{-1}A^{-1}) = I\). Similarly, we can show that \((B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}(I)B = B^{-1}B = I\). Since multiplying \((AB)\) by \((B^{-1}A^{-1})\) in both orders results in the identity matrix, \((B^{-1}A^{-1})\) is indeed the inverse of \((AB)\).
Analyzing the Options for \((AB)^{-1}\)
The question provides four options for what \((AB)^{-1}\) is equal to:
- B\(^{-1}\)A\(^{-1}\)
- A\(^{-1}\)B\(^{-1}\)
- B\(^{-1}\)A
- A\(^{-1}\)B
Based on the property of the inverse of a matrix product that we derived and verified, \((AB)^{-1}\) is equal to \(B^{-1}A^{-1}\).
Comparing this with the given options:
- Option 1: B\(^{-1}\)A\(^{-1}\) - This matches our derived result.
- Option 2: A\(^{-1}\)B\(^{-1}\) - This is the product of the inverses in the original order, which is generally incorrect because matrix multiplication is not commutative (AB is not necessarily equal to BA, so \((A^{-1}B^{-1})(AB)\) is generally not I).
- Option 3: B\(^{-1}\)A - This involves the original matrix A, not its inverse.
- Option 4: A\(^{-1}\)B - This involves the original matrix B, not its inverse.
Therefore, the correct expression for \((AB)^{-1}\) is \(B^{-1}A^{-1}\).
Conclusion
For two invertible square matrices A and B of the same order, the inverse of their product \((AB)\) is given by the product of their individual inverses in reverse order, which is \(B^{-1}A^{-1}\).
Revision Table: Key Matrix Properties
| Property | Description | Formula |
|---|---|---|
| Inverse of a Product | The inverse of the product of two invertible matrices is the product of their inverses in reverse order. | \((AB)^{-1} = B^{-1}A^{-1}\) |
| Inverse of an Inverse | The inverse of the inverse of a matrix is the original matrix itself. | \((A^{-1})^{-1} = A\) |
| Inverse of a Transpose | The inverse of the transpose of a matrix is the transpose of its inverse. | \((A^T)^{-1} = (A^{-1})^T\) |
| Inverse of Scalar Multiple | The inverse of a scalar multiple of a matrix (where k is a non-zero scalar) | \((kA)^{-1} = \frac{1}{k}A^{-1}\) |
| Associativity of Multiplication | Matrix multiplication is associative. | \(A(BC) = (AB)C\) |
Additional Information on Matrix Inverses
- Existence of Inverse: A square matrix A has an inverse if and only if its determinant is non-zero (\(\det(A) \neq 0\)). If the determinant is zero, the matrix is called singular, and it does not have an inverse.
- Uniqueness of Inverse: If an inverse exists for a matrix A, it is unique. There is only one matrix \(A^{-1}\) that satisfies \(AA^{-1} = A^{-1}A = I\).
- Non-Commutativity: In general, matrix multiplication is not commutative, meaning \(AB \neq BA\). This is why the order matters when taking the inverse of a product: \((AB)^{-1} = B^{-1}A^{-1}\) and not \(A^{-1}B^{-1}\).
- Identity Matrix: The identity matrix \(I\) acts like the number 1 in matrix multiplication. For any matrix A (of compatible size), \(AI = IA = A\). The identity matrix is always square and has 1s on the main diagonal and 0s elsewhere.
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