# 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:

Solution:

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

Solution:

## 3.2 Matrix arithmetic

### Practice Problems

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

a)
b)
c)
d)
e)
f)

Solutions:

a)

b)

c)

d)

e)

f)

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

a)

Solution:

b)

Solution:

c)

Solution:

d)

Solution:

e)

Solution:

f)

Solution:

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

Solution: Yes.

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

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.

Solution: Yes.

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

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

### External Resources

Khan Academy: Intro to Matrix Multiplication

Khan Academy: Is Matrix Multiplication Commutative?

Khan Academy: Associative Property of Matrix Multiplication

## 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?

Solution:

Neither.

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

Solution:

Symmetric.

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

Solution:

Neither.

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

Solution:

Skew.

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

Solution:

Symmetric.

Problem 3.3.6: Consider the following matrices:

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.

Solution: Yes.

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

Solution: No.

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

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)
b)
c)
d)
e)
f)

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.

Solution: -18

Problem 3.4.3: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.

Solution: -9

Problem 3.4.4: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.

Solution: 54

Problem 3.4.5: Find the determinant of the following 3×3 matrix using both Laplace expansion and the diagonalization shortcut.

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!

Solution: 23

Problem 3.4.7: Verify

using the following matrices:

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!

Solution: Yes. (Determinant = 2)

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

Solution: No. (Determinant = 0)

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

Solution: No. (Determinant = 0)

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

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

## 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).

Solution:

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

Solution:

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

Solution:

Not invertible.

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

Solution:

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

Solution:

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

Solution: x=-3, y=1

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

Solution: No solutions exist.

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

Solution: x=3, y=-2

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

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

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

Solution: No solution exists.

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

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.

Solution: x=-3, y=1

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

Solution: No solution exists.

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

Solution: x=3, y=-2

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

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

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

Solution: No solution exists.

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

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

### External Resources

No external resources for this section.