Matrices and Determinants - Definition, Difference, Examples
Matrices and Determinants - Definition, Difference, Examples
TL;DR
Matrices and determinants are two different objects: a matrix is a rectangular grid of numbers, while a determinant is a single number you compute from a square matrix. This article covers both definitions, the exact difference between them, how to find a 2x2 and 3x3 determinant, the properties that make them useful, and six worked examples.
What Are Matrices and Determinants?
A matrix is a rectangular arrangement of numbers set out in rows and columns and written inside square brackets. A determinant is a single scalar value calculated from a square matrix, one number that summarises the whole grid. The matrix is the container; the determinant is one fact about it.
Here is a 2x2 matrix and its determinant side by side. The matrix:
A=[ \begin{bmatrix} 3 & 1 \ 2 & 4 \end{bmatrix} ]
Its determinant, written det(A) or |A|:
|A|=(3)(4)−(1)(2)=10
What Is the Difference Between a Matrix and a Determinant?
Three differences matter:
A matrix is a grid; a determinant is a number. You cannot "equal" a matrix to a number, but a determinant is a number, so |A| = 10 is a complete statement.
A matrix can be any shape; a determinant needs a square matrix. A 2×3 matrix is fine, but it has no determinant. Determinants only exist for n×n matrices.
Multiplying by a constant behaves differently. Multiply a matrix by k and every entry is multiplied. Multiply a determinant by k and only the entries of one row (or one column) get multiplied.
| Feature | Matrix | Determinant |
|---|---|---|
| What it is | A grid of numbers | A single number |
| Notation | Square brackets [ ] | Vertical bars $ |
| Shape allowed | Any m×n | Square n×n only |
| Multiply by k | Scales every entry | Scales one row or column |
| Main use | Stores and transforms data | Tests invertibility, solves systems |
Examples of Matrices and Determinants
Example 1
Find the determinant of [ \begin{bmatrix} 5 & 2 \ 1 & 3 \end{bmatrix} ].
Apply the 2x2 rule |A|=ad−bc:
|A|=(5)(3)−(2)(1)
|A|=15−2
|A|=13
Final answer: |A|=13.
Example 2
Find the determinant of [ \begin{bmatrix} -4 & 6 \ -2 & 3 \end{bmatrix} ].
Wrong attempt. |A| = (−4)(3)+(6)(−2)=−12−12=−24.
Why it is wrong. The determinant subtracts the off-diagonal product.
Correct. |A|=(−4)(3)−(6)(−2):
|A|=−12−(−12)
|A|=0
Final answer: |A|=0.
Example 3
Multiply [ \begin{bmatrix} 1 & 2 \ 0 & 3 \end{bmatrix} ] by [ \begin{bmatrix} 4 & 1 \ 2 & 5 \end{bmatrix} ], then find the determinant of the result.
Multiply row-by-column:
[ \begin{bmatrix} 1 & 2 \ 0 & 3 \end{bmatrix} \begin{bmatrix} 4 & 1 \ 2 & 5 \end{bmatrix} = \begin{bmatrix} 8 & 11 \ 6 & 15 \end{bmatrix} ].
Now find the determinant:
|AB|=(8)(15)−(11)(6)
|AB|=120−66
|AB|=54
Final answer: |AB|=54.
Example 4
Find the determinant of the 3x3 matrix [ \begin{bmatrix} 2 & 1 & 3 \ 0 & 4 & 1 \ 5 & 2 & 1 \end{bmatrix} ].
|A|=2|\begin{bmatrix} 4 & 1 \ 2 & 1 \end{bmatrix}|−1|\begin{bmatrix} 0 & 1 \ 5 & 1 \end{bmatrix}|+3|\begin{bmatrix} 0 & 4 \ 5 & 2 \end{bmatrix}|
|A|=2(4−2)−1(0−5)+3(0−20)
|A|=2(2)−1(−5)+3(−20)
|A|=4+5−60
|A|=−51
Final answer: |A|=−51.
Example 5
A scalar multiple. Given |A|=7 for a 3x3 matrix, find |2A|.
|2A|=2^3|A|
|2A|=8×7
|2A|=56
Final answer: |2A|=56.
Example 6
Use a determinant to test invertibility. Is [ \begin{bmatrix} 6 & 3 \ 4 & 2 \end{bmatrix} ] invertible?
|A|=(6)(2)−(3)(4)
|A|=0
Final answer: Not invertible.
Why Do Matrices and Determinants Matter? "One number that decides everything"
Matrices exist to handle complex problems like connected circuits or chemical reactions. A matrix holds the system, while a determinant answers a key question.
Solving linear systems. Cramer's rule uses determinants to directly solve for each unknown.
Computer graphics. Transformations in graphics depend on matrix multiplication; determinants ensure correct orientation.
Quantum mechanics. Matrix mechanics described atomic physics using matrices.
Properties of Matrices and Determinants You Will Reuse
Matrix properties. Addition is commutative and associative. Multiplication is associative but not commutative. The transpose swaps rows and columns.
Determinant properties.
- Swapping two rows flips the sign.
- A row of zeros or two identical rows makes |A|=0.
- |AB|=|A| |B|.
- |AT|=|A|.
- |A| for a triangular matrix is the product of diagonal entries.
Where Students Trip Up on Matrices and Determinants
Mistakes often arise with matrix notation and determinant calculations. Pay attention to signs and dimensions.
The Mathematicians Behind Matrices and Determinants
Arthur Cayley, James Joseph Sylvester, and Gottfried Wilhelm Leibniz made significant contributions to the field.
Key Takeaways
- Matrices are grids; determinants are single numbers.
- Only square matrices have determinants.
- The determinant indicates invertibility.
Practice Questions on Matrices and Determinants
- Find the determinant of [ \begin{bmatrix} 7 & 4 \ 2 & 3 \end{bmatrix} ].
- Find the determinant of [ \begin{bmatrix} 2 & -1 & 0 \ 1 & 3 & 4 \ 0 & 2 & 1 \end{bmatrix} ].
- Given |A|=5, find |3A|.
- Is [ \begin{bmatrix} 8 & 4 \ 6 & 3 \end{bmatrix} ] invertible?
- If |A|=4 and |B|=2, what is |AB|?
Answer to Question 1: 13. Answer to Question 2: -9. Answer to Question 3: 135. Answer to Question 4: Not invertible. Answer to Question 5: 8.