Static Equilibrium Analysis: Using Matrix Algebra

J. Zachary Klingensmith

3 Static Equilibrium Analysis: Using Matrix Algebra

3.1 An Introduction to matrices

Practice Problems

Problem 3.1.1: Write the following system of equations as a set of matrices:

$3x-y+2z=4$
$-x+4y+9z=-7$
$-4x-y+z=5$

Solution:

$\begin{bmatrix}3 & -1 & 2 \\ -1 & 4 & 9 \\ -4 & -1 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z\end{bmatrix}=\begin{bmatrix} 4 \\ -7 \\ 5\end{bmatrix}$

Problem 3.1.2: Write the following system of equations as a set of matrices:

$2t-5u-6v+2w=4$
$-t+u+2v-w=6$
$5t+2u+4w=1$
$3u-v+w=0$

Solution:

$\begin{bmatrix} 2 & -5 & -6 & 2 \\ -1 & 1 & 2 & -1 \\ 5 & 2 & 0 & 4 \\ 0 & 3 & -1 & 1 \end{bmatrix} \begin{bmatrix} t \\ u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 4 \\ 6 \\ 1 \\ 0\end{bmatrix}$

External Resources

Khan Academy: Introduction to Matrices

3.2 Matrix arithmetic

Practice Problems

Problem 3.2.1: Conduct each of the operations below given the following matrices:

$A=\begin{bmatrix}3 & 1 & 0 \\ 2 & -3 & -1\\0 & 6 & 2\end{bmatrix}, B=\begin{bmatrix}0 & -2 & 3\\5 & 1 & 4\\2 & 0 & -5\end{bmatrix}, C=\begin{bmatrix} -4 & 2 & -5\\0 & -3 & 2\\2 & 4 & -3\end{bmatrix}$

a) $A+C$
b) $2B$
c) $A+B+C$
d) $2C-A$
e) $4C-2B$
f) $3(B-A)$

Solutions:

a) $\begin{bmatrix} 3 & -1 & 3 \\ 7 & -2 & 3 \\ 2 & 6 & -3\end{bmatrix}$

b) $\begin{bmatrix} 0 & -4 & 6 \\ 10 & 2 & 8 \\ 4 & 0 & -10 \end{bmatrix}$

c) $\begin{bmatrix} -1 & 1 & -2 \\ 7 & -5 & 5 \\ 4 & -10 & 6 \end{bmatrix}$

d) $\begin{bmatrix} -11 & 3 & -10 \\ -2 & -3 & 5 \\ 4 & 2 & -8 \end{bmatrix}$

e) $\begin{bmatrix} -16 & 12 & -26 \\ -10 & -14 & 0 \\ 4 & 16 & -2 \end{bmatrix}$

f) $\begin{bmatrix} -9 & -9 & 3 \\ 9 & 12 & 15 \\ 6 & -18 & -21 \end{bmatrix}$

Problem 3.2.2: Conduct each of the operations below given the following matrices (the same as in problem 3.2.1):

$A=\begin{bmatrix}3 & 1 & 0 \\ 2 & -3 & -1\\0 & 6 & 2\end{bmatrix}, B=\begin{bmatrix}0 & -2 & 3\\5 & 1 & 4\\2 & 0 & -5\end{bmatrix}, C=\begin{bmatrix} -4 & 2 & -5\\0 & -3 & 2\\2 & 4 & -3\end{bmatrix}$

a) $AC$

Solution:

$\begin{bmatrix} -12 & 3 & -13 \\ -10 & -9 & -1 \\ 4 & -10 & 6 \end{bmatrix}$

b) $BC$

Solution:

$\begin{bmatrix} 6 & 18 & -13 \\ -12 & 23 & -35 \\ -18 & -16 & 5 \end{bmatrix}$

c) $2BC$

Solution:

$\begin{bmatrix} 12 & 36 & -26 \\ -24 & 46 & -70 \\ -36 & -32 & 10 \end{bmatrix}$

d) $CB$

Solution:

$\begin{bmatrix} 0 & 10 & 21 \\ -11 & -3 & -22 \\ 14 & 0 & 37 \end{bmatrix}$

e) $2A(B+C)$

Solution:

$\begin{bmatrix} -14 & -4 & 0 \\ -54 & 4 & -28 \\ 76 & -8 & 40 \end{bmatrix}$

f) $C(2A-B)$

Solution:

$\begin{bmatrix} -16 & -90 & 45 \\ -1 & 45 & 36 \\ 14 & -56 & -57 \end{bmatrix}$

Problem 3.2.3: Can we multiply the following matrices? If not, why? If so, find the product.

$\begin{bmatrix}1 & 0 & 3 & -1\end{bmatrix}\times\begin{bmatrix}2 \\ 1 \\ 3 \\ 6\end{bmatrix}$

Solution: Yes.

$\begin{bmatrix} 5 \end{bmatrix}$

Problem 3.2.4: Can we multiply the following matrices? If not, why? If so, find the product.

$\begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix}\times\begin{bmatrix} 0 & 3 \\ 2 & -1 \\ 3 & 3\end{bmatrix}$

Solution: No. (2 columns for the first matrix and 3 rows for the second matrix.)

Problem 3.2.5: Can we multiply the following matrices? If not, why? If so, find the product.

$\begin{bmatrix} 3 & 2 & 4 & 0 \\ 0 & 1 & -3 & -2 \end{bmatrix}\times\begin{bmatrix} 4 \\ 1 \\ 2 \\ -1\end{bmatrix}$

Solution: Yes.

$\begin{bmatrix} 22 \\ -3 \end{bmatrix}$

Problem 3.2.6: Can we multiply the following matrices? If not, why? If so, find the product.

$\begin{bmatrix} 0 \\ 3 \\ -2\end{bmatrix} \times \begin{bmatrix} 2 \\ 3 \\ -1\end{bmatrix}$

Solution: No. (1 column for the first matrix and 3 rows for the second matrix.)

External Resources

Khan Academy: Adding and Subtracting Matrices

Khan Academy: Scalar Multiplication

Khan Academy: Intro to Matrix Multiplication

Khan Academy: Multiplying Matrices

Khan Academy: Is Matrix Multiplication Commutative?

Khan Academy: Associative Property of Matrix Multiplication

Khan Academy: Zero Matrix

3.3 The transposE, identity, and inverse matrices

Practice Problems

Problem 3.3.1: Find the transpose of the following matrix. Is it symmetric, skew, or neither?

$\begin{bmatrix} 3 & 0 & 1\\1 & 0 & 3\\0 & 3 & 1\end{bmatrix}$

Solution:

$\begin{bmatrix} 3 & 1 & 0 \\ 0 & 0 & 3 \\ 1 & 3 & 1\end{bmatrix}$

Neither.

Problem 3.3.2: Find the transpose of the following matrix. Is it symmetric, skew, or neither?

$\begin{bmatrix} 3 & -2 & 4\\-2 & 6 & 2\\4 & 2 & 3\end{bmatrix}$

Solution:

$\begin{bmatrix} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3\end{bmatrix}$

Symmetric.

Problem 3.3.3: Find the transpose of the following matrix. Is it symmetric, skew, or neither?

$\begin{bmatrix} 2 & 1 & 3 & 4\\3 & -1 & -2 & 2\\0 & 1 & 3 & 2\end{bmatrix}$

Solution:

$\begin{bmatrix} 2 & 3 & 0 \\ 1 & -1 & 1 \\ 3 & -2 & 3 \\ 4 & 2 & 2\end{bmatrix}$

Neither.

Problem 3.3.4: Find the transpose of the following matrix. Is it symmetric, skew, or neither?

$\begin{bmatrix} 0 & 1 & -2\\-1 & 0 & 3\\2 & -3 & 0\end{bmatrix}$

Solution:

$\begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -3 \\ -2 & 3 & 0\end{bmatrix}$

Skew.

Problem 3.3.5: Find the transpose of the following matrix. Is it symmetric, skew, or neither?

$\begin{bmatrix} 3 & -1 & 0 & 4\\-1 & 2 & 1 & 5\\0 & 1 & -4 & -2\\4 & 5 & -2 & 6\end{bmatrix}$

Solution:

$\begin{bmatrix} 3 & -1 & 0 & 4 \\ -1 & 2 & 1 & 5 \\ 0 & 1 & -4 & -2 \\ 4 & 5 & -2 & 6\end{bmatrix}$

Symmetric.

Problem 3.3.6: Consider the following matrices:

$A=\begin{bmatrix} 1 & 0 & 3\\2 & -2 & 3\end{bmatrix}, B=\begin{bmatrix}1 & 3 & 2 & 0\end{bmatrix}, C=\begin{bmatrix}3 & 0\\2 & -1\end{bmatrix}.$

For each, determine what identity matrix is needed for both left-side multiplication (ex: IA) and right-side multiplication (ex: AI). Show that the identity matrix works.

Solution: For AI, use the 3×3 identity matrix. For IA, use the 2×2 identity matrix. For BI, use a 4×4 identity matrix. For IB, use a 1×1 identity matrix. For CI, use a 2×2 identity matrix. For IC, use a 2×2 identity matrix.

Problem 3.3.7: Determine if matrices A and B are inverses of each other.

$A=\begin{bmatrix}2 & 1\\0&1\end{bmatrix} B=\begin{bmatrix}1/2 & -1/2 \\ 0 & 1\end{bmatrix}$

Solution: Yes.

Problem 3.3.8: Determine if matrices A and B are inverses of each other.

$A=\begin{bmatrix}3 & 0\\1 & 2\end{bmatrix} B=\begin{bmatrix}0 & 3\\0 &2\end{bmatrix}$

Solution: No.

Problem 3.3.9: Determine if matrices A and B are inverses of each other.

$A=\begin{bmatrix}0 & 1 & 1\\1 & 1 & 1\\0 & 0& 1\end{bmatrix} B=\begin{bmatrix}-1 & 1 & 0\\1 & 0 & -1\\0 & 0 & 1\end{bmatrix}$

Solution: Yes.

External Resources

Khan Academy: Introduction to the Identity Matrix

Khan Academy: Dimensions of an Identity Matrix

Khan Academy: Using Identity and Zero Matrices

3.4 determinants: An Introduction

Practice Problems

Problem 3.4.1: Find the determinant of the following 2×2 matrices.

a) $\begin{bmatrix} 2 & -2 \\ 3 & 3\end{bmatrix}$
b) $\begin{bmatrix}5 & -3 \\ 1 & 6\end{bmatrix}$
c) $\begin{bmatrix} 0 & -2\\3 & 5\end{bmatrix}$
d) $\begin{bmatrix} -6 & -4 \\ 2 & 5\end{bmatrix}$
e) $\begin{bmatrix} 1 & -5 \\ 9 & 2 \end{bmatrix}$
f) $\begin{bmatrix} -4 & 7 \\ -2 & 5\end{bmatrix}$

Solutions: 12, 33 ,6, -22, 47, -6

Problem 3.4.2: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.

$\begin{bmatrix} 1 & 5 & 2\\3 & 1 & 1\\0 & 2 & 2\end{bmatrix}$

Solution: -18

Problem 3.4.3: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.
$\begin{bmatrix} 1 & 4 & -2\\1 & 1 & 0\\0 & 3 & 1\end{bmatrix}$

Solution: -9

Problem 3.4.4: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.
$\begin{bmatrix} 5 & 3 & 1\\ -2 & 4 & 2\\ 1 & 0 & 2\end{bmatrix}$

Solution: 54

Problem 3.4.5: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.
$\begin{bmatrix} 4 & 2 & 0\\ 0 & 2 & 0 \\ 3 & 1 & -1\end{bmatrix}$

Solution: -8

Problem 3.4.6: Find the determinant of this 4×4 matrix. I will not ask you to do something like this on an exam!

$\begin{bmatrix} 3 & 0 & 1 & 1 \\ -1 & 3 & 2 & 2\\ 0 & -3 & 4 & 1\\ 1 & 2 & -1 & 0\end{bmatrix}$

Solution: 23

Problem 3.4.7: Verify
$det(A)\times det(B)=det(AB)$

using the following matrices:

$A=\begin{bmatrix} 3 & 1\\-1 & 4\end{bmatrix}, B=\begin{bmatrix} 2 & 0 \\1& 1\end{bmatrix}$

Solution: Determinant is 26 (13×2)

Problem 3.4.8: Does the following matrix have an inverse? If it does…DO NOT FIND THE INVERSE!

$\begin{bmatrix} 2 & 4 \\ -1 & -1\end{bmatrix}$

Solution: Yes. (Determinant = 2)

Problem 3.4.9: Does the following matrix have an inverse? If it does…DO NOT FIND THE INVERSE!

$\begin{bmatrix} 1 & 5\\2 & 10\end{bmatrix}$

Solution: No. (Determinant = 0)

Problem 3.4.10: Does the following matrix have an inverse? If it does…DO NOT FIND THE INVERSE!

$\begin{bmatrix} 3 & 0 & 2\\1 & -3 & 2\\9 & 0 & 6\end{bmatrix}$

Solution: No. (Determinant = 0)

Problem 3.4.11: Does the following matrix have an inverse? If it does…DO NOT FIND THE INVERSE!

$\begin{bmatrix} 1 & 3 & 3\\1 & 3 & 0\\1 & 0 &3\end{bmatrix}$

Solution: Yes. (Determinant = -9)

External Resources

Khan Academy: Determinant of a 2×2 Matrix

Khan Academy: Determinant of a 3×3 Matrix: Standard Method (Laplace Expansion)

Khan Academy: Determinant of a 3×3 Matrix: Shortcut Method

Khan Academy: Intro to Matrix Inverses

Khan Academy: Determining Invertible Matrices

3.5 Determinants: Finding the inverse

Practice Problems

Problem 3.5.1: Find the inverse of the following matrix, if it exists, using Laplace Expansion (using the adjoint matrix).

$\begin{bmatrix} 3 & 1 \\ 1 & 2\end{bmatrix}$

Solution:

$\begin{bmatrix} 2/5 & -1/5 \\ -1/5 & 3/5 \end{bmatrix}$

Problem 3.5.2: Find the inverse of the following matrix, if it exists, using Laplace Expansion (using the adjoint matrix).

$\begin{bmatrix} 5 & 0 \\ 3 & -1\end{bmatrix}$

Solution:

$\begin{bmatrix} 1/5 & 0 \\ 3/5 & -1\end{bmatrix}$

Problem 3.5.3: Find the inverse of the following matrix using Laplace Expansion (using the adjoint matrix).

$\begin{bmatrix} 4 & 2 \\ 2 & 1\end{bmatrix}$

Solution:

Not invertible.

Problem 3.5.4: Find the inverse of the following matrix using Laplace Expansion (using the adjoint matrix).

$\begin{bmatrix} 2 & 0 & 2\\ 1 & 1 & 1\\ 0 & 3 & 1\end{bmatrix}$

Solution:

$\begin{bmatrix} -1 & 3 & -1 \\ -1/2 & 1 & 0 \\ 3/2 & -3 & 1 \end{bmatrix}$

Problem 3.5.5: Find the inverse of the following matrix using Laplace Expansion (using the adjoint matrix).

$\begin{bmatrix} -1 & 2 & 4 \\ 0 & 1 & 1\\3 & 1 & 1\end{bmatrix}$

Solution:

$\begin{bmatrix} 0 & -1/3 & 1/3 \\ -1/2 & 13/6 & -1/6 \\ 1/2 & -7/6 & 1/6\end{bmatrix}$

Problem 3.5.6: Find the solution of the following system of equations using Laplace Expansion.

$4x+2y=-10$
$3x+9y=0$

Solution: x=-3, y=1

Problem 3.5.7: Find the solution of the following system of equations using Laplace Expansion.

$-x+2y=-1$
$5x-10y=6$

Solution: No solutions exist.

Problem 3.5.8: Find the solution of the following system of equations using Laplace Expansion.

$4x+2y=8$
$5x+3y=9$

Solution: x=3, y=-2

Problem 3.5.9: Find the solution of the following system of equations using Laplace Expansion.

$5x-2y+3z=20$
$2x-4y-3z=-9$
$x+6y-8z=21$

Solution: x=5, y=4, z=1

Problem 3.5.10: Find the solution of the following system of equations using Laplace Expansion.

$2x+3y-6z=1$
$-4x-6y+12z=-2$
$x+2y+5z=10$

Solution: No solution exists.

Problem 3.5.11: Find the solution of the following system of equations using Laplace Expansion.

$2x-y-z=3$
$x+y=-1$
$3x-y-2z=7$

Solution: x=-1, y=0, z=-5

External Resources

Khan Academy: Finding 2×2 Inverse Using Determinant

Khan Academy: Solving Linear Systems with Matrix Equations

3.6 Cramer’s Rule

Practice Problems

Problem 3.6.1: Find the solution to the following system of equations using Cramer’s Rule.

$4x+2y=-10$
$3x+9y=0$

Solution: x=-3, y=1

Problem 3.6.2: Find the solution to the following system of equations using Cramer’s Rule.

$-x+2y=-1$
$5x-10y=6$

Solution: No solution exists.

Problem 3.6.3: Find the solution to the following system of equations using Cramer’s Rule.

$4x+2y=8$
$5x+3y=9$

Solution: x=3, y=-2

Problem 3.6.4: Find the solution to the following system of equations using Cramer’s Rule.

$5x-2y+3z=20$
$2x-4y-3z=-9$
$x+6y-8z=21$

Solution: x=5, y=4, z=1

Problem 3.6.5: Find the solution to the following system of equations using Cramer’s Rule.

$2x+3y-6z=1$
$-4x-6y+12z=-2$
$x+2y+5z=10$

Solution: No solution exists.

Problem 3.6.6: Find the solution to the following system of equations using Cramer’s Rule.

$2x-y-z=3$
$x+y=-1$
$3x-y-2z=7$

Solution: x=-1, y=0, z=-5

External Resources

No external resources for this section.

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Student Companion for Mathematical Economics Copyright © by J. Zachary Klingensmith is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

3.1 An Introduction to matrices

Practice Problems

External Resources

3.2 Matrix arithmetic

Practice Problems

External Resources

3.3 The transposE, identity, and inverse matrices

Practice Problems

External Resources

3.4 determinants: An Introduction

Practice Problems

External Resources

3.5 Determinants: Finding the inverse

Practice Problems

External Resources

3.6 Cramer’s Rule

Practice Problems

External Resources

License

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