Spring 2026: Math 291 Homework

Any page and section numbers in the assignments below refer to Heffron's text.

Tuesday, January 20

1. Verify properties 1-8 from today's lecture, for \(A = \begin{pmatrix} 1 & -9\\2 & 6\end{pmatrix}\), \(B = \begin{pmatrix} 1 & 4\\0 & -9\end{pmatrix}\), \(C = \begin{pmatrix}0 & -4\\9 & 2\end{pmatrix}\), \(\lambda = 7, \lambda_1 = -6, \lambda_2 = 4\).

2. Give a proof of the cancellation property using entries in the matrices rather than the proof given class.

Thursday, January 22

1. For the matrices \(A, B, C\) in the previous assignment, verify:

  1. (i) \(A(B+C) = AB+AC\).
  2. (ii) \(A(BC) = (AB)C\).

2. For the matrix \(A = \begin{pmatrix} 3 & 1\\5 & 2\end{pmatrix}\) first verify that \(A^{-1} = \begin{pmatrix} 2 & -1\\-5 & 3\end{pmatrix}\), and then use \(A^{-1}\) to solve the system of equations

\[\begin{align*} 3x+y &= 7\\ 5x+2y &= -3. \end{align*}\]

3. Use mathematical induction to prove the following statements:

  1. (i) \(1^2+2^2+3^2+\cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\), for all \(n\geq 1\).
  2. (ii) \(9^n-1\) is divisible by 8, for all \(n\geq 1\).
Tuesday, January 27

1. For \(A = \begin{pmatrix} 1 & -9\\2 & 6\end{pmatrix}\), \(B = \begin{pmatrix} 1 & 4\\0 & -9\end{pmatrix}\) and \(\lambda = 7\), verify properties 1-5 of the determinant given in today's lecture.

2. For the \(2\times 2\) matrix \(A\), we verified in class that if \(\textrm{det}(A)\not = 0\), then \(A^{-1}\) exists. Prove that if \(A^{-1}\) exists, then \(\det A \not = 0\). Thus, we have the following:

Theorem. The \(2\times 2\) matrix is invertible if and only if \(\det A\not = 0\).

We'll see later in the semester that this holds for any \(n\times n\) matrix.

3. Here are three systems of linear equations. Identify which one has a unique solution, infinitely many solutions and no solutions.

System A
\[\begin{align*} 2x + 3y &= 7 \\ 6x + 9y &= 31 \end{align*}\]
System B
\[\begin{align*} 2x + 3y &= -1 \\ 6x + 2y &= 4 \end{align*}\]
System C
\[\begin{align*} 2x + 3y &= 7 \\ 6x + 9y &= 21 \end{align*}\]
Thursday, January 29

1. For the three systems of equations given in the previous assignment, use augmented matrices and Gaussian elimination to find the solution set of each system.

2. Something new: Find the solution set to the system of equations below using Gaussian elimination, following as closely as you can the algorithm given in class. Hint: You'll have to introduce a parameter to describe the solution set.

\[\begin{align*} 2x+4y+6z &= 12\\ x+y+z &= 8. \end{align*}\]

3. Suppose that the ordered pair \((s,t)\) is a solution to the system

\[\begin{align*} ax+by &= u\\ cx+dy &= v. \end{align*}\]

Verify that \((s,t)\) is a solution to each of systems below. Assume that \(\lambda \in \mathbb{R}\) and is non-zero for System C.

System A
\[\begin{align*} cx + dy &= v \\ ax + by &= u \end{align*}\]
System B
\[\begin{align*} ax + by &= u \\ (c+\lambda a)x + (d+\lambda b)y &=v+\lambda u \end{align*}\]
System C
\[\begin{align*} ax + by &= u \\ \lambda cx + \lambda dy &= \lambda v \end{align*}\]

Now assume that \((s,t)\) is a solution to A, B, or C, and show that \((s,t)\) is a solution to the original system of equations. You must consider all three cases. These calculations show that solutions to systems of equations are invariant under elementary row operations.