How to Find Determinants of a Matrix?
For every square matrix, you can calculate the determinant of the matrix. Here is a step-by-step guide to finding the determinants of a matrix.
Tutor-style math help
Find Determinants of a Matrix: what to notice and how to work it
Matrices skill
Matrix work starts with dimensions. The rows and columns tell you whether an operation is allowed before you calculate anything.
What to notice first
Write the matrix size first. Addition, multiplication, determinants, and inverses all have dimension rules.
Common student mistake
Do not multiply matrices entry by entry. Matrix multiplication uses each row of the first matrix with each column of the second.
Key formulas and cues
((mtimes n)+(mtimes n)text{ is allowed})
((mtimes n)(ntimes p)=mtimes p)
(detbegin{pmatrix}a&bc&dend{pmatrix}=ad-bc)
(A^{-1}A=I)
A reliable path
- Check dimensionsRows by columns determines what operation is legal.
- Use the correct ruleAddition is entry-by-entry; multiplication is row-by-column.
- Interpret the resultFor systems, translate the matrix answer back into variables.
Worked examples
Add matrices
Example: ([1 4]+[2 3])
- The matrices have the same size.
- Add matching entries.
- Compute each position.
Answer: ([3 7])
Multiplication size
Example: 2 by 3 matrix times 3 by 2 matrix
- Inner dimensions match: 3 and 3.
- The product is allowed.
- Outer dimensions give the result size.
Answer: 2 by 2 matrix
Try one before moving on
Try: What is the size of a 4×2 matrix times a 2×5 matrix?
Answer: 4×5.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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Find Determinants of a Matrix: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Can you add a 2×2 matrix and a 3×2 matrix?
2. A 2×3 matrix times a 3×4 matrix gives:
3. A determinant is defined for:
A Matrix is an array of numbers: (m×n) with (m) rows and (n) columns. The determinant of a matrix is a scalar value that is defined for square matrices.
Related Topics
A step-by-step guide to finding determinants of a matrix
- The determinant of a (2×2) matrix (A), (A=begin{bmatrix}a & b c & d end{bmatrix}) is defined as (|A|=ad-bc).
- The determinant of a (3×3) matrix (A), (A=begin{bmatrix}a & b & c d & e & f g & h & iend{bmatrix}) is defined as (|A|=a(ei-fh)-b(di-fg)+c(dh-eg))
Finding Determinants of a Matrix – Example 1:
Evaluate the determinant of matrix (A=begin{bmatrix}5 & -1 6 & 2 end{bmatrix})
Solution:
The determinant is: (|A|=5(2)-(-1)(6)=10-(-6)=10+6=16)
Finding Determinants of a Matrix – Example 2:
Evaluate the determinant of matrix: \(A=\begin{bmatrix}2 & 0 & 1 \\ 0 & -1 & 1 \\ 3 & 1 & -2\end{bmatrix}\)
Original price was: $27.99.$17.99Current price is: $17.99.
Solution:
The determinant is: 2[(-1)(-2) – (1)(1)] + 1[(0)(1) – (-1)(3)] = 2(1) + 3 = 5
Exercises for Finding Determinants of a matrix
Evaluate the determinants of each matrix.
- \(\begin{bmatrix}3 & 5 \\ 0 & 9\end{bmatrix}\)
- (color{blue}{begin{bmatrix}0 & 1 4 & 6 end{bmatrix}})
- \(\begin{bmatrix}1 & 5 & 4 \\ 0 & 9 & 1 \\ 1 & 0 & 6\end{bmatrix}\)
- (color{blue}{begin{bmatrix}6 & 0 & 51 & 4 & 23 & 7 & 4end{bmatrix}})
- (color{blue}{27})
- (color{blue}{-4})
- (color{blue}{23})
- (color{blue}{-13})
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