Problem 59 - Discrete Mathematics with Applications

Question

Prove each. If \(A\) and \(B\) are two invertible matrices of order \(n,\) then \((A B)^{-1}=B^{-1} A^{-1} .\)

Step-by-Step Solution

Answer

Let A and B be two invertible matrices of order n. To prove (AB)^(-1) = B^(-1)A^(-1), we multiply AB by B^(-1)A^(-1): 1. \((AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AA^{-1} = I\) Similarly, we multiply B^(-1)A^(-1) by AB: 2. \((B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}IB = B^{-1}B = I\) Since both products result in the identity matrix, we can conclude that (AB)^(-1) = B^(-1)A^(-1).

Step 1: State the given information

Let A and B be two invertible matrices of order n.

Step 2: Define inverse of a matrix

Recall that a matrix M is invertible if there exists a matrix M^(-1) such that MM^(-1)=M^(-1)M=I, where I is the identity matrix of the same order as M.

Step 3: Show that (AB)(B^(-1)A^(-1)) is the identity matrix

We need to prove that (AB)^(-1)=B^(-1)A^(-1). To do this, we will multiply AB by B^(-1)A^(-1). Note that we are allowed to do this because matrix multiplication is associative. So, we get: \( (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1}\) Since B and its inverse B^(-1) multiply to the identity matrix, we can simplify the equation: \(A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1}\) Now, using the properties of the inverse matrix, we have: \( AA^{-1} = I\)

Step 4: Show that (B^(-1)A^(-1))(AB) is the identity matrix

Similarly, we can multiply B^(-1)A^(-1) by AB and simplify: \( (B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}IB = B^{-1}B \) Since B and its inverse B^(-1) multiply to the identity matrix, we get: \( B^{-1}B = I\)

Step 5: Conclusion

Since both (AB)(B^(-1)A^(-1)) and (B^(-1)A^(-1))(AB) result in the identity matrix, we can conclude that (AB)^(-1) = B^(-1)A^(-1).

Key Concepts

Invertible Matrices

Invertible matrices are special types of matrices that possess an inverse. The concept is similar to the inverse of a number, such as how multiplying any number by its inverse results in 1. For a matrix, the inverse is another matrix that, when multiplied together, yields the identity matrix. This property makes invertible matrices extremely important in linear algebra.
To dive a bit deeper:

• A matrix is said to be invertible or non-singular if it has a square shape, meaning the same number of rows and columns.
• If you have a matrix \(M\), its inverse is often denoted as \(M^{-1}\).
• The criterion for \(M\) being invertible is that \(MM^{-1} = I\), where \(I\) is the identity matrix.

“I” in this context is not just any letter but represents a special matrix that acts like the number 1 in matrix multiplication.

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix. Unlike adding or subtracting matrices, matrix multiplication involves combining rows and columns in a specific way, which even allows for the mixing of matrices of different dimensions under certain rules. Understanding how to properly multiply matrices is a key skill in mathematics:

• The number of columns in the first matrix must equal the number of rows in the second matrix to perform multiplication.
• If you have matrices \(A\) and \(B\), and you multiply them to get \(C = AB\), \(C\) usually has different values than \(BA\) unless they are special kinds of matrices.
• Matrix multiplication is associative, so \(A(BC) = (AB)C\), which is a versatile property in matrix operations.

Consider matrix multiplication akin to mixing paint colors, where the order can affect the final result.

Identity Matrix

The identity matrix acts like the number 1 in matrix operations. It has a property known as multiplicative identity, which means that when any matrix is multiplied by an identity matrix, it remains unchanged. This makes the identity matrix a crucial component in the study of matrices:

• The identity matrix is always square-shaped, having the same number of rows and columns.
• Along its main diagonal, from the top left to the bottom right, the components are 1’s, and all other components are zero.
• In notation, an identity matrix is represented by \(I\). For example, a 2x2 identity matrix looks like \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

The identity matrix effectively doesn’t change a matrix when used in multiplication, preserving the original matrix characteristics.