Problem 47 - Algebra 2

Question

CHALLENGE For which values of \(a, b, c,\) and \(d\) will \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]=A^{-1} ?\)

Step-by-Step Solution

Answer

Not possible with given condition.

Step 1: Understand the Problem

We need to determine the values of \(a\), \(b\), \(c\), and \(d\) such that matrix \(A\) is equal to its inverse. This occurs when \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix.

Step 2: Determine the Condition for the Inverse

A 2x2 matrix \(A\) has an inverse if its determinant is not zero. The determinant of \(A\) is \(ad - bc\). Thus, we need \(ad - bc eq 0\).

Key Concepts

2x2 Matrix

A 2x2 matrix is simply a square matrix with two rows and two columns. This means it is composed of four elements arranged in a grid. For example:

• First row: elements \(a\) and \(b\)
• Second row: elements \(c\) and \(d\)

The structure of a 2x2 matrix is typically represented as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]This type of matrix is basic yet significant in linear algebra because it forms the building block of larger matrices and is crucial for operations such as addition, subtraction, and more importantly, multiplication. Understanding these simple matrices is critical to grasp more complex linear algebra concepts.

Determinant

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix like:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant is calculated using the formula \(ad - bc\).

• It is crucial because a matrix only has an inverse if its determinant is non-zero.
• A zero determinant indicates that the matrix cannot be inverted, meaning the matrix is singular.

Hence, if you want to find the inverse of a matrix, checking the determinant is the first step. This determinant also gives insight into the scaling factor of the transformation represented by the matrix, making it a valuable tool in understanding matrix properties.

Identity Matrix

The identity matrix is a special kind of matrix in which all the elements in the leading diagonal are ones, and all other elements are zeros. For a 2x2 identity matrix:\[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

• The identity matrix acts like "1" in matrix multiplication. It doesn't change the other matrix it is multiplied with.
• Multiplying any matrix by the identity matrix results in the same matrix.

In the context of matrix inversion, finding the inverse of a matrix \(A\) and multiplying it by \(A\) should yield the identity matrix, symbolizing the idea of reverting an operation.

Inverse Matrix

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. For a 2x2 matrix \(A\) with elements \(a, b, c,\) and \(d\), it has an inverse if and only if the determinant \(ad - bc\) is not zero.

• The formula to find the inverse is given by:\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]
• It is vital to remember that not all matrices have inverses. Only those with a non-zero determinant are invertible.
• The inverse matrix is analogous to the reciprocal of numbers in arithmetic.

In practical applications, finding an inverse is useful for solving systems of linear equations, among other things, as it allows us to "undo" the effect of the matrix.