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Chapter 7: Problem 9

If the matrices $$ \left[\begin{array}{rr} 2 & -1 \\ 3 & a \end{array}\right] \text { and }\left[\begin{array}{rr} 2 & b \\ 3 & -4 \end{array}\right] $$ are singular, find the values of \(a\) and \(b\).

Short Answer

Expert verified
The values are \(a = -\frac{3}{2}\) and \(b = -\frac{8}{3}\).

Step by step solution

01

Define Singular Matrix

A matrix is singular if its determinant is zero. Therefore, to find the values of \(a\) and \(b\), calculate the determinants of the given matrices and set them to zero.
02

Calculate Determinant of First Matrix

Compute the determinant of the first matrix \[ \begin{vmatrix} 2 & -1 \ 3 & a \ \ \ \end{vmatrix} = 2a - (-1) \times 3 = 2a + 3 \] Set the determinant equal to zero to find \(a\): \[ 2a + 3 = 0 \ \ 2a = -3 \ \ a = -\frac{3}{2} \] Therefore, \(a = -\frac{3}{2}\).
03

Calculate Determinant of Second Matrix

Compute the determinant of the second matrix \[ \begin{vmatrix} 2 & b \ 3 & -4 \ \ \ \end{vmatrix} = 2(-4) - (b)(3) = -8 - 3b \] Set the determinant equal to zero to find \(b\): \[ -8 - 3b = 0 \ \ -3b = 8 \ \ b = -\frac{8}{3} \] Therefore, \(b = -\frac{8}{3}\).

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Key Concepts

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

Determinant
The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant is found using a simple formula. The formula for a matrix \(\begin{bmatrix} a & b \ c & d \ \end{bmatrix}\) is given by: \(\text{det} = ad - bc\). The determinant helps us understand whether a matrix is singular or non-singular. If the determinant is zero, the matrix is singular, meaning it doesn't have an inverse. Understanding determinants is crucial in linear algebra because they play a key role in matrix operations and solving equations.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It involves the study of matrices and systems of linear equations, which are essential for solving real-world problems. In linear algebra, understanding concepts like singular and non-singular matrices, determinants, and matrix operations is fundamental. These concepts form the backbone for more advanced topics in mathematics, physics, engineering, and computer science.
Matrix Operations
Matrix operations include addition, subtraction, multiplication, and finding the inverse of matrices. These operations are essential tools in linear algebra. For example, to determine if a matrix is singular or non-singular, we calculate its determinant. Singular matrices are special because they don't have an inverse. Other operations like matrix addition and multiplication follow specific rules. Understanding these operations allows us to manipulate matrices effectively and solve complex equations.
Solving Equations
Solving equations with matrices often involves using determinants and matrix operations. For a system of linear equations represented as \(\textbf{A}\textbf{x} = \textbf{b}\), where \(\textbf{A}\) is a matrix, \(\textbf{x}\) is a vector of variables, and \(\textbf{b}\) is a vector of constants, we need to determine if \(\textbf{A}\) is singular. If it's singular (determinant is zero), we can't find a unique solution. In such cases, the system may have infinitely many solutions or none at all. Understanding these concepts helps us solve systems of equations more efficiently.

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Most popular questions from this chapter

Find (where possible) the inverse of the matrices $$ \mathbf{A}=\left[\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 3 & 2 \\ -1 & 2 & 1 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{rrr} 1 & 4 & 5 \\ 2 & 1 & 3 \\ -1 & 3 & 2 \end{array}\right] $$ Are these matrices singular or non-singular?

Each unit of engineering output requires as input \(0.2\) units of engineering and \(0.4\) units of transport. Each unit of transport output requires as input \(0.2\) units of engineering and \(0.1\) units of transport. Determine the level of total output needed to satisfy a final demand of 760 units of engineering and 420 units of transport.

Consider the macroeconomic model defined by $$ \begin{array}{ll} Y=C+I^{*}+G^{*}+X^{*}-M & \\ C=a Y+b & (00) \\ M=m Y+M^{*} & \left(00\right) \end{array} $$ Show that this system can be written as \(\mathrm{Ax}=\mathrm{b}\), where $$ \mathbf{A}=\left[\begin{array}{ccc} 1 & -1 & 1 \\ -a & 1 & 0 \\ -m & 0 & 1 \end{array}\right] \mathbf{x}=\left[\begin{array}{c} Y \\ C \\ M \end{array}\right] \quad \mathbf{b}=\left[\begin{array}{c} I^{*}+G^{*}+X^{*} \\ b \\ M^{*} \end{array}\right] $$ Use Cramer's rule to show that $$ Y=\frac{b+I^{*}+G^{*}+X^{*}-M^{*}}{1-a+m} $$ Write down the autonomous investment multiplier for \(Y\) and deduce that \(Y\) increases as as \(/^{*}\) increases.

The general linear supply and demand equations for a one-commodity market model are given by $$ \begin{array}{ll} P=a Q_{\mathrm{s}}+b & (a>0, b>0) \\ P=-c Q_{\mathrm{D}}+d & (c>0, d>0) \end{array} $$ Show that in matrix notation the equilibrium price, \(P\), and quantity, \(Q\), satisfy $$ \left[\begin{array}{cc} 1 & -a \\ 1 & c \end{array}\right]\left[\begin{array}{l} P \\ Q \end{array}\right]=\left[\begin{array}{l} b \\ d \end{array}\right] $$ Solve this system to express \(P\) and \(Q\) in terms of \(a, b, c\) and \(d\). Write down the multiplier for \(Q\) due to changes in \(b\) and deduce that an increase in \(b\) leads to a decrease in \(Q\).

Verify the equations (a) \(\mathbf{A}(\mathbf{B}+\mathbf{C})=\mathbf{A} \mathbf{B}+\mathbf{A C}\) (b) \((\mathbf{A B}) \mathrm{C}=\mathbf{A}(\mathbf{B C})\) in the case when $$ \mathbf{A}=\left[\begin{array}{rr} 5 & -3 \\ 2 & 1 \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{ll} 1 & 5 \\ 4 & 0 \end{array}\right] \text { and } \quad \mathbf{C}=\left[\begin{array}{rr} -1 & 1 \\ 1 & 2 \end{array}\right] $$
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