Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

If \(A\) is a \(2 \times 2\) matrix given by \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) then \(A\) is invertible if and only if \(a d-b c \neq 0 .\) If \(a d-b c \neq 0,\) verify that the inverse is \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\).

Step-by-Step Solution

Answer

The solution to verifying the inverse of the provided \(2 \times 2\) matrix \(A\) involves confirming the invertibility condition \(ad - bc \neq 0\). Then, form the supposed inverse using the formula and confirm it by multiplying with the original matrix \(A\) to ensure it produces the Identity Matrix. If it does, then the purported inverse is indeed the correct inverse.

Step 1: Confirm the invertibility condition

To ensure that the matrix \(A\) is invertible, it's necessary to confirm whether the determinant of the matrix \(A\) does not equal to zero. That means, \(ad - bc \neq 0\) given the matrix \(A\).

Step 2: Write down the form of inverse

Following the formula for inverse of 2x2 matrix, the inverse matrix \(A^{-1}\) should be \(\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \ -c & a\end{array}\right]\) if \(A\) is invertible.

Step 3: Demonstrate the multiplication

To prove the correctness of determined inverse, multiply matrix \(A\) with its inverse \(A^{-1}\), in both orders, because matrix multiplication is not generally commutative. The result of the multiplication should yield the Identity matrix in both cases, as this is a primary property of matrix inverses. For \(A \times A^{-1}\) and \(A^{-1} \times A\), both should equal to the \(2 \times 2\) Identity Matrix \(\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]\). This shows that the given 'inverse' of matrix \(A\) is indeed its true inverse.

Key Concepts

Matrix Determinant

The determinant of a matrix is a special number that can help us to understand many properties of the matrix, such as whether it is invertible. For a 2x2 matrix, which is structured like this:

• Matrix A: \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]

The determinant can be calculated using the formula:

• Determinant of A: \[det(A) = ad - bc\]

This computation is vital because if the determinant is not zero, the matrix has an inverse. However, when the determinant equals zero, it means that the matrix is singular and does not have an inverse. This connection between the determinant and the invertibility of a matrix is crucial when delving into linear algebra.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of 2 rows and 2 columns. Each entry in the matrix can be represented by a notation that relates it to its position in the matrix, usually like this:

• Matrix A:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]

Here, \(a\), \(b\), \(c\), and \(d\) are specific numbers that fill the spaces in this 2x2 formation. Understanding the structure and arrangement of the matrix is essential, as it serves as the basis for calculating its determinant and forming its inverse. Such matrices are widely used due to their simplicity in both academic exercises and real-world applications like computer graphics and physics.

Identity Matrix

An Identity Matrix acts like the number 1 in regular arithmetic, serving as the neutral element in matrix multiplication. For a 2x2 matrix, the identity matrix is expressed as:

• Identity Matrix I: \[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

When you multiply any matrix by an identity matrix, the original matrix remains unchanged. This property extends to the concept of inverses, where multiplying a matrix by its inverse results in the identity matrix. This means:

• For matrix \(A\): \(A \times A^{-1} = I\) and \(A^{-1} \times A = I\)

Recognizing and using the identity matrix is key to validating whether a calculated inverse is correct.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce another matrix. This operation is pivotal when verifying the inverse of a matrix. The procedure for multiplying a 2x2 matrix by another 2x2 matrix involves taking the dot product of rows and columns.Suppose we multiply matrices \(A\) and \(B\):

• Matrix A: \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]
• Matrix B: \[B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\]

The resulting matrix \(C\) is calculated as:

• First row of \(C\): \([ae + bg, af + bh]\)
• Second row of \(C\): \([ce + dg, cf + dh]\)

This description highlights that the order of multiplication matters because the dot product could result in different values. Practicing matrix multiplication is crucial for tasks like proving that a matrix product equals the identity matrix when dealing with inverses.