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Chapter 9: Problem 16

Find the inverse, if it exists, for each matrix. $$\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Short Answer

Expert verified
The matrix is not invertible as its determinant is zero.

Step by step solution

01

Calculate the Determinant

To determine if the matrix \[ \begin{bmatrix} -6 & 4 \ -3 & 2 \ \ \ \right] \] is invertible, first find its determinant using the formula for a 2x2 matrix: \( \text{det}(A) = ad - bc \). Here, \( a = -6 \), \( b = 4 \), \( c = -3 \), and \( d = 2 \): \( \text{det}(A) = (-6 \times 2) - (4 \times -3) \) Simplify the equation: \( \text{det}(A) = -12 + 12 = 0 \)
02

Analyze the Determinant

Since \( \text{det}(A) = 0 \), the matrix does not have an inverse. A matrix is only invertible if its determinant is non-zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant
The determinant of a matrix is a special number that can tell us a lot about the matrix. For a 2x2 matrix, the determinant is computed using the formula: \ \( \text{det}(A) = ad - bc \) \ where the matrix has the form: \ \[ \begin{pmatrix} a & b \ c & d \ \right] \] \
The determinant helps determine whether a matrix is invertible. If the determinant equals zero, the matrix has no inverse (it is called singular or non-invertible). This is because we cannot divide by zero in mathematics, and the formula for finding an inverse involves dividing by the determinant.
Invertible Matrix
An invertible matrix is one that has an inverse. To check if a matrix is invertible, calculate its determinant:
  • If the determinant is not zero (\text{det}(A) 鈮� 0), then the matrix is invertible.
  • If the determinant is zero (\text{det}(A) = 0), then the matrix is not invertible.

For a 2x2 matrix \( A \), if it is invertible, the inverse can be found using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \ \right] \]
However, if the determinant is zero, the formula for the inverse cannot be used and the matrix does not have an inverse.
2x2 Matrix
A 2x2 matrix is a basic type of matrix that has two rows and two columns. It has the form: \[ \begin{pmatrix} a & b \ c & d \ \right] \]
  • The first row contains the elements \( a \) and \( b \).
  • The second row contains the elements \( c \) and \( d \).

The 2x2 matrix is fundamental in linear algebra and is often used to introduce the concept of matrix operations and properties, such as determinants, inverses, and systems of linear equations. It is a good starting point for understanding more complex matrices and their applications in various fields.

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Most popular questions from this chapter

Find the equation of the circle passing through the given points. $$(-5,0),(2,-1), \text { and }(4,3)$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}$$

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Solve each problem. At the Brendan Berger ranch, 6 goats and 5 sheep sell for \(\$ 305,\) while 2 goats and 9 sheep sell for \(\$ 285 .\) Find the cost of a single goat and of a single sheep.

Investment Decisions Jane Hooker invests \(40,000\) received as an inheritance in three parts. With one part she buys mutual funds that offer a return of \(2 \%\) per year. The second part, which amounts to twice the first, is used to buy government bonds paying \(2.5 \%\) per year. She puts the rest of the money into a savings account that pays \(1.25 \%\) annual interest. During the first year, the total interest is \(\$ 825 .\) How much did she invest at each rate?
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