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Determinants

Session Material:

Lay: 3.1-3.3

Recap and Exercises

Session Notes

Session Material

Session Description

This session introduces the concept of the determinant of a matrix. We will start by defining and learning how to compute the determinant for \(2 \times 2\) matrices, and then extend this to \(3 \times 3\) matrices and \(n \times n\) matrices, using a systematic method.

We will explore the basic properties of the determinant, including how row operations affect its value. A central result that will be highlighted is the relationship between the determinant and a matrix's invertibility – a matrix is invertible if and only if its determinant is non-zero.

Depending on the precise content of the sections, we may also touch upon the geometric interpretation of the determinant, such as its connection to area or volume.

Key Concepts

  • Determinant
  • Calculating 2x2 Determinants
  • Calculating 3x3 Determinants
  • Properties of Determinants
  • Determinant and Invertibility
  • Geometric Interpretation (Area/Volume)

Learning Objectives

  • Compute determinants of \(2 \times 2\), \(3 \times 3\), and \(n \times n\) matrices using systematic methods.
  • Apply properties of determinants to simplify calculations and understand matrix behavior.
  • Relate the determinant to matrix invertibility and solve related problems.
  • Interpret the geometric meaning of determinants in terms of area and volume.
  • Analyze the effect of row operations on the determinant.

Exercises

Exercise 1 (3.1.1-3.1.2)

Compute the determinants using a cofactor expansion across the first row. Then compute the determinant also by a cofactor expansion down the second column.

  1. \(\left|\begin{array}{rrr}3 & 0 & 4 \\ 2 & 3 & 2 \\ 0 & 5 & -1\end{array}\right|\)
  2. \(\left|\begin{array}{rrr}0 & 5 & 1 \\ 4 & -3 & 0 \\ 2 & 4 & 1\end{array}\right|\)
 
  1. 1
  2. 2

Exercise 2 (3.1.9)

Compute the determinant by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.

$\displaystyle \left|\begin{array}{rrrr} 6 & 0 & 0 & 5 \\ 1 & 7 & 2 & -5 \\ 2 & 0 & 0 & 0 \\ 8 & 3 & 1 & 8 \end{array}\right|$
 

10

Exercise 3 (3.1.43)

\([\mathbf{M}]\) Is it true that \(\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B ?\) To find out, generate random \(5 \times 5\) matrices \(A\) and \(B\), and compute \(\operatorname{det}(A+B)-\operatorname{det} A-\operatorname{det} B\). (Refer to Exercise 37 in Section 2.1.) Repeat the calculations for three other pairs of \(n \times n\) matrices, for various values of \(n\). Report your results.

 

Here are sample results testing whether \(\det(A+B)=\det A+\det B\) for random integer matrices \(A,B\) of sizes \(n=5,2,3,10\):

n det(A) det(B) det(A + B) det(A + B) − det(A) − det(B)
5 668 −1077 2765 3174
2 −22 0 −26 −4
3 −93 28 −27 38
10 3.11 × 10⁶ 3.35 × 10⁷ 3.28 × 10⁸ 2.92 × 10⁸

In every case, \(\det(A+B)-\det A-\det B\neq0\), demonstrating that \(\det(A+B)\neq\det A+\det B\). In general, determinant is not an additive function of matrices.

Exercise 4 (3.2.1-3.2.2)

Both equations illustrate a property of determinants. State the property.

  1. \(\left|\begin{array}{rrr}0 & 5 & -2 \\ 1 & -3 & 6 \\ 4 & -1 & 8\end{array}\right|=-\left|\begin{array}{rrr}1 & -3 & 6 \\ 0 & 5 & -2 \\ 4 & -1 & 8\end{array}\right|\)
  2. \(\left|\begin{array}{rrr}2 & -6 & 4 \\ 3 & 5 & -2 \\ 1 & 6 & 3\end{array}\right|=2\left|\begin{array}{rrr}1 & -3 & 2 \\ 3 & 5 & -2 \\ 1 & 6 & 3\end{array}\right|\)
 
  1. Rows 1 and 2 are interchanged, so the determinant changes sign (Theorem 3b.).
  2. The constant 2 may be factored out of the Row 1 (Theorem 3c.).

Exercise 5 (3.2.15-3.2.20)

Find the determinants in the following exercises, where

\[ \left|\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right|=7 . \]
  1. \(\left|\begin{array}{ccc}a & b & c \\ d & e & f \\ 5 g & 5 h & 5 i\end{array}\right|\)
  2. \(\left|\begin{array}{ccc}a & b & c \\ 3 d & 3 e & 3 f \\ g & h & i\end{array}\right|\)
  3. \(\left|\begin{array}{lll}a & b & c \\ g & h & i \\ d & e & f\end{array}\right|\)
  4. \(\left|\begin{array}{lll}g & h & i \\ a & b & c \\ d & e & f\end{array}\right|\)
  5. \(\left|\begin{array}{ccc}a & b & c \\ 2 d+a & 2 e+b & 2 f+c \\ g & h & i\end{array}\right|\)
  6. \(\left|\begin{array}{ccc}a+d & b+e & c+f \\ d & e & f \\ g & h & i\end{array}\right|\)
 
  1. 35
  2. 21
  3. -7
  4. 7
  5. 14
  6. 7

Exercise 6 (3.2.25)

Use the determinant to decide if the set of vectors is linearly independent.

$\left[\begin{array}{r}7 \\ -4 \\ -6\end{array}\right],\left[\begin{array}{r}-8 \\ 5 \\ 7\end{array}\right],\left[\begin{array}{r}7 \\ 0 \\ -5\end{array}\right]$
 

The columns of the matrix form a linearly independent set.