Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Matrix Entry, find the value of the constant \(k\) such that \(B=A^{-1}\). $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 0 \end{array}\right], \quad B=\left[\begin{array}{rr} k & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{4} \end{array}\right]$$

Step-by-Step Solution

Answer

The value of the constant k is 0

Step 1: Calculate the inverse of A

To find the inverse of any matrix A, A should first be a square matrix. Let's assume that A is a 2x2 matrix such that A=\[\begin{array}{rr} a & b\ c & d \end{array}\right]. Then the inverse of A, A^{-1} will be equal to 1/(ad-bc) \[\begin{array}{rr} d & -b\ -c & a \end{array}\right]. Then, the matrix A is \[\begin{array}{rr} 1 & 2 \ -2 & 0 \end{array}\right], thus, the inverse of matrix A will be equal to 1/(1*0 - (-2*2)) \[\begin{array}{rr} 0 & -2\ 2 & 1 \end{array}\right]

Step 2: Equate the inverse of matrix A to matrix B

Given that B=A^{-1}, we then have B=\[\begin{array}{rr} 0 & -2\ 2 & 1 \end{array}\right], which is supposed to equal to \[\begin{array}{rr}k & -\frac{1}{2}\\frac{1}{2} & \frac{1}{4}\end{array}\right] by equating these two matrices, we find that k=0.

Step 3: Cross-check the Answer

Substituting 'k' into the original matrix B, we get \[\begin{array}{rr} 0 & -\frac{1}{2}\ \frac{1}{2} & \frac{1}{4} \end{array}\right]. This is equivalent to A^{-1}, hence the answer checks out.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that is calculated from a square matrix. It is crucial for determining whether a matrix has an inverse or not. The determinant helps us understand if a matrix can be inverted, which is essential in solving matrix equations.
In a 2x2 matrix, the determinant can be easily calculated using the formula:\[ \text{det}(A) = ad - bc \] where \( A \) is the matrix \( \begin{array}{cc} a & b \ c & d \end{array} \).
If the determinant is zero, the matrix does not have an inverse, meaning it is singular.
For matrix \( A \) given as \( \begin{array}{cc} 1 & 2 \ -2 & 0 \end{array} \), we compute its determinant as follows: \( (1)(0) - (-2)(2) = 0 + 4 = 4 \).
This non-zero determinant indicates that the matrix \( A \) is invertible. Thus, we can proceed to find its inverse.

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix that contains 2 rows and 2 columns. These matrices are particularly straightforward to work with in matrix algebra.
Every 2x2 matrix has the form:

• \( \begin{array}{cc} a & b \ c & d \end{array} \)

This simplicity allows for easier calculation of concepts like the determinant and finding the inverse.
The process for finding the inverse of such a matrix is quite straightforward, once the determinant is known. The inverse of a 2x2 matrix \( A \), \( A^{-1} \), is calculated with the formula:\[ A^{-1} = \frac{1}{ad-bc} \begin{array}{cc} d & -b \ -c & a \end{array} \]
This demonstrates how each element of the matrix is adjusted to create the inverse matrix, which undoes the effect of the original matrix when multiplied together, provided its determinant is non-zero.

Matrix Equations

Matrix equations involve expressions that contain matrices and their operations, such as addition, subtraction, and multiplication. Solving matrix equations often involves finding the inverse of matrices.
For instance, given \( B = A^{-1} \), we are tasked to equate two matrices, the inverse of matrix \( A \) and matrix \( B \).
Finding \( A^{-1} \) and setting it equal to \( B \) revealed the unknown constant \( k \). This type of comparison is essential in understanding how matrices interact.
Matrix equations can also represent complex systems in various fields like physics or engineering, making them an excellent tool in applied mathematics.
Solving these equations is often a matter of substituting known values and solving for the unknowns, much like solving traditional algebraic equations.
This is critical in fields where systems of equations need simplifying or solving within larger problems.