Question

If \(\mathrm{A}=\left[\begin{array}{lr}3 & -2 \\ 6 & 4\end{array}\right]\), then \(\mathrm{AA}^{-1}=\)____. (1) \(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) (2) \(\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]\) (3) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (4) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\)

Short answer

Question: Find the product of matrix A and its inverse, where \(A = \begin{bmatrix} 3 & -2 \\ 6 & 4 \end{bmatrix}\). Choose the correct answer from the following options: (1) \(\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]\) (2) \(\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\) (3) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (4) \(\left[\begin{array}{ll}3 & -2 \\ 6 & 4\end{array}\right]\) Answer: (3) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\)

Step-by-step solution

Find the inverse of matrix A

To find the inverse of a 2x2 matrix, we will follow these steps: 1. Find the determinant of the matrix: det(A) = ad - bc 2. Swap a and d 3. Change the signs of b and c 4. Multiply each element by the reciprocal of the determinant For the given matrix A, \(A = \begin{bmatrix} 3 & -2 \\ 6 & 4 \end{bmatrix}\) det(A) = (3 * 4) - (-2 * 6) = 12 + 12 = 24 Now, perform steps 2-4: Swap a and d: $ \begin{bmatrix} 4 & -2 \\ 6 & 3 \end{bmatrix} $ Change the signs of b and c: $ \begin{bmatrix} 4 & 2 \\ -6 & 3 \end{bmatrix} $ And multiply each element by the reciprocal of the determinant (1/24): $ A^{-1} = \frac{1}{24} \begin{bmatrix} 4 & 2 \\ -6 & 3 \end{bmatrix} = \begin{bmatrix} \frac{1}{6} & \frac{1}{12} \\ -\frac{1}{4} & \frac{1}{8} \end{bmatrix} $

Multiply matrix A by its inverse

Now that we have the inverse of matrix A, we need to multiply A by its inverse. For matrix multiplication, follow the row-by-column method: $ AA^{-1} = \begin{bmatrix} 3 & -2 \\ 6 & 4 \end{bmatrix} \begin{bmatrix} \frac{1}{6} & \frac{1}{12} \\ -\frac{1}{4} & \frac{1}{8} \end{bmatrix} = \begin{bmatrix} (3 \cdot \frac{1}{6}) + (-2 \cdot -\frac{1}{4}) & (3 \cdot \frac{1}{12}) + (-2 \cdot \frac{1}{8}) \\ (6 \cdot \frac{1}{6}) + (4 \cdot -\frac{1}{4}) & (6 \cdot \frac{1}{12}) + (4 \cdot \frac{1}{8}) \end{bmatrix} = \begin{bmatrix} \frac{1}{2} + \frac{1}{2} & \frac{1}{4} - \frac{1}{4} \\ 1 - 1 & \frac{1}{2} + \frac{1}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $ So, \(\mathrm{AA}^{-1} = \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\). The correct answer is (3).

Key concepts

Matrix Inverse

When delving into the realm of linear algebra, the concept of a matrix inverse is crucial for solving various types of problems. A matrix inverse is, in many ways, similar to the reciprocal of a number. Just like multiplying a number by its reciprocal gives 1, when you multiply a matrix by its inverse, you get the identity matrix.

Understanding the inverse of a matrix starts with recognizing that not all matrices have inverses. A matrix must be square (same number of rows and columns) and its determinant, which we'll discuss next, cannot be zero. To calculate the inverse of a 2x2 matrix, you must perform a series of steps, including finding and using the determinant, swapping elements, and changing signs as found in the step by step solution provided.

When you multiply a matrix by its inverse, the result should be the identity matrix, confirming the accuracy of your calculations. Involving matrix inversion in system-solving, finding determinants, and working with eigenvalues are just a few of the instances where this powerful tool is utilized.

Determinant of a Matrix

The determinant of a matrix is an essential value which can be calculated from its elements. In computational terms, it's akin to a 'numeric signature' for square matrices. For a 2x2 matrix, the formula is quite simple: \( det(A) = ad - bc \) where \( a, b, c, d \) are the elements of the matrix positioned as \( a \) and \( d \) on the diagonal from the top left to the bottom right, and \( b \) and \( c \) are the off-diagonal elements.

The determinant serves multiple purposes: determining if a matrix is invertible (a non-zero determinant), influencing the volume change caused by a linear transformation, and even playing a role in solving systems of linear equations. For matrices larger than 2x2, the calculation becomes more complex, involving minors and cofactors, but the principle remains the same. While the step by step solution provided makes it look straightforward, remember the significance of the determinant extends far beyond just a step in finding a matrix inverse.

Identity Matrix

In the context of matrix operations, the identity matrix plays a role analogous to the number 1 in arithmetic. It is a square matrix that retains the original matrix when multiplied with it, hence the term 'identity'. The defining feature of the identity matrix, denoted as \( I \) or sometimes \( E \) depending on the context, is that all of the diagonal elements are 1s, and all other elements are 0s. This property provides the scaffold for many matrix operations including matrix inversion.

If you multiply any matrix by the identity matrix of appropriate dimensions, the result will be the original matrix. This is comparable to multiplying any number by 1, which results in the same number. The solution in this exercise rightly concludes with the identity matrix when a matrix is multiplied by its inverse, demonstrating a fundamental and expected outcome in linear algebra.