Question

Matrices \(A\) and \(B\) are given. Compute \((A B)^{-1}\) and \(B^{-1} A^{-1}\). $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{ll} 7 & 1 \\ 2 & 1 \end{array}\right] $$

Short answer

\((AB)^{-1} = B^{-1}A^{-1} = \begin{bmatrix} -0.7 & 0.3 \\ 2.9 & -1.1 \end{bmatrix}\)

Step-by-step solution

Compute the Product AB

To find the inverse of the product \((AB)^{-1}\), we first need to calculate \(AB\). Multiply matrices \(A\) and \(B\):\[AB = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\begin{bmatrix} 7 & 1 \ 2 & 1 \end{bmatrix} = \begin{bmatrix} (1\cdot7 + 2\cdot2) & (1\cdot1 + 2\cdot1) \ (3\cdot7 + 4\cdot2) & (3\cdot1 + 4\cdot1) \end{bmatrix} = \begin{bmatrix} 11 & 3 \ 29 & 7 \end{bmatrix}\]

Calculate Inverse of Matrix AB

Next, we compute the inverse of the product \(AB\), which is \((AB)^{-1}\). We use the formula for the inverse of a 2x2 matrix:\[(AB)^{-1} = \frac{1}{\text{det}(AB)}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Where \(a=11\), \(b=3\), \(c=29\), and \(d=7\). First, calculate the determinant:\[\text{det}(AB) = 11\cdot7 - 3\cdot29 = 77 - 87 = -10\]So, the inverse is:\[(AB)^{-1} = \frac{1}{-10}\begin{bmatrix} 7 & -3 \ -29 & 11 \end{bmatrix} = \begin{bmatrix} -0.7 & 0.3 \ 2.9 & -1.1 \end{bmatrix}\]

Compute Inverse of A

Now compute the inverse of matrix \(A\) using its determinant:\[\text{det}(A) = 1\cdot4 - 2\cdot3 = 4 - 6 = -2\]Then, \[A^{-1} = \frac{1}{-2}\begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix}\]

Compute Inverse of B

Compute the inverse of matrix \(B\):\[\text{det}(B) = 7\cdot1 - 1\cdot2 = 7 - 2 = 5\]Then, \[B^{-1} = \frac{1}{5}\begin{bmatrix} 1 & -1 \ -2 & 7 \end{bmatrix} = \begin{bmatrix} 0.2 & -0.2 \ -0.4 & 1.4 \end{bmatrix}\]

Compute \(B^{-1}A^{-1}\)

Multiply the inverses of \(B\) and \(A\):\[B^{-1}A^{-1} = \begin{bmatrix} 0.2 & -0.2 \ -0.4 & 1.4 \end{bmatrix}\begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} = \begin{bmatrix} (0.2\cdot(-2) + (-0.2)\cdot1.5) & (0.2\cdot1 + (-0.2)\cdot(-0.5)) \ (-0.4\cdot(-2) + 1.4\cdot1.5) & (-0.4\cdot1 + 1.4\cdot(-0.5)) \end{bmatrix}\]Simplifying the calculations yields:\[B^{-1}A^{-1} = \begin{bmatrix} -0.7 & 0.3 \ 2.9 & -1.1 \end{bmatrix}\]

Conclusion: Compare (AB)^{-1} and B^{-1}A^{-1}

The results for both calculations, \((AB)^{-1}\) and \(B^{-1}A^{-1}\), are equal:\[(AB)^{-1} = B^{-1}A^{-1} = \begin{bmatrix} -0.7 & 0.3 \ 2.9 & -1.1 \end{bmatrix}\]This demonstrates that for any invertible matrices \(A\) and \(B\) of the same size, \((AB)^{-1} = B^{-1}A^{-1}\).

Key concepts

Matrix Multiplication

When we talk about matrix multiplication, we're referring to the process of multiplying two matrices together to yield a third matrix. This is not the same as multiplying ordinary numbers, where you simply multiply corresponding pairs and add them; it's a bit more complex.
Here's how it generally works: for two matrices, say Matrix A and Matrix B, the resulting matrix is obtained by taking each row element of A and multiplying it with each column element of B. Then, sum these products.

• Ensure the number of columns in the first matrix matches the number of rows in the second matrix to perform multiplication.
• The element in the resulting matrix is found by summing the products obtained from multiplying each element of a row of the first matrix by each corresponding element of a column of the second matrix.
• This new matrix is called the product of the two matrices.

For example, when matrices: \[A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 7 & 1 \ 2 & 1 \end{bmatrix}\]are multiplied, the resulting matrix is:\[AB = \begin{bmatrix} 11 & 3 \ 29 & 7 \end{bmatrix}\] It provides us with a new way to pivot information contained by both matrices.

Inverse of a Matrix

When dealing with matrices, the concept of an inverse is quite parallel to the notion of a reciprocal of a number. Specifically, for a matrix to have an inverse, it must be square (same number of rows and columns) and non-singular (determinant not equal to zero).
The inverse of a matrix 'A' is denoted as \(A^{-1}\), and the product of a matrix and its inverse gives the identity matrix, much like multiplying a number by its reciprocal results in one.
Here's how to find it for a 2x2 matrix:

• Given a matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the inverse is expressed as \[A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]
• The condition to invertibility is that the determinant (\(ad - bc\)) should not be zero.
• This inverse allows 'undoing' the effect of the original matrix operation.
• It is crucial in solving systems of linear equations where matrices are involved.

Practically, inverses are used to navigate backward through operations represented by the original matrix.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It's an essential value especially in the field of linear algebra, as it helps determine whether a matrix has an inverse or not. If the determinant is zero, the matrix does not have an inverse (then it’s called singular).
For a 2x2 matrix such as \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant is calculated as:

• Calculated as \(ad - bc\).
• This value signifies whether a 2x2 matrix is invertible or not.
• A non-zero value means the matrix can be inverted, as was demonstrated in the exercise.
• Determining the determinant is a straightforward yet powerful tool in matrix algebra that plays a vital role in various computations.

Determinants transform matrices into single scalar values, simplifying decisions about matrix properties.

Properties of Matrix Inverses

Understanding the properties of matrix inverses deepens comprehension of their utility, especially when dealing more complex computations involving multiple matrices.

• A major rule is that the inverse of a product of two matrices equals the product of their inverses in reverse order, expressed as: \[(AB)^{-1} = B^{-1} A^{-1}.\]
• This means when you have matrices A and B, their product’s inverse can be found by individually inverting them and multiplying them in reverse.
• The identity matrix is unique in that it is its own inverse; multiplying a matrix by its inverse results in an identity.
• Not all matrices are invertible; only nonsingular ones with a non-zero determinant can have an inverse.

An intuitive understanding of these properties simplifies solving systems of equations, cryptographic algorithms, and aids in theoretical proofs within linear algebra.