# Quick Math Determinant

## Quick Math Determinant

1
##### Enter expression in this form

Example (2x2 Matrix):  Enter expression as [2, 5], [11, 7]

### Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, such as whether the matrix is invertible and information about the matrix’s linear transformations, including scaling factors and volume changes. The determinant is denoted by $$\text{det}(A)$$ or $$|A|$$, where $$A$$ is the matrix.

#### Key Properties of Determinants

1. Invertibility: A matrix $$A$$ is invertible if and only if $$\text{det}(A) \neq 0$$.
2. Multiplicative Property: $$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$$ for any two square matrices $$A$$ and $$B$$.
3. Transpose: $$\text{det}(A^T) = \text{det}(A)$$.
4. Effect of Row Operations:
– Swapping two rows multiplies the determinant by -1.
– Multiplying a row by a scalar $$k$$ multiplies the determinant by $$k$$.
– Adding a multiple of one row to another row does not change the determinant.

#### Calculation of Determinants

For small matrices, determinants can be calculated directly using specific formulas.

Determinant of a 2×2 Matrix

For a 2×2 matrix $$A$$ given by:
$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

The determinant is:
$\text{det}(A) = ad – bc$

Determinant of a 3×3 Matrix

For a 3×3 matrix $$A$$ given by:
$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$

The determinant is:
$\text{det}(A) = a(ei – fh) – b(di – fg) + c(dh – eg)$

#### Worked Examples

##### Example 1: Determinant of a 2×2 Matrix

Matrix:
$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Solution:
$\text{det}(A) = (1 \cdot 4) – (2 \cdot 3) = 4 – 6 = -2$

Answer: $$\text{det}(A) = -2$$.

##### Example 2: Determinant of a 3×3 Matrix

Matrix:
$A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$

Solution:
$\text{det}(A) = 1 \cdot (1 \cdot 0 – 4 \cdot 6) – 2 \cdot (0 \cdot 0 – 4 \cdot 5) + 3 \cdot (0 \cdot 6 – 1 \cdot 5)$
$= 1 \cdot (0 – 24) – 2 \cdot (0 – 20) + 3 \cdot (0 – 5)$
$= 1 \cdot (-24) – 2 \cdot (-20) + 3 \cdot (-5)$
$= -24 + 40 – 15$
$= 1$

Answer: $$\text{det}(A) = 1$$.

##### Example 3: Determinant of a 4×4 Matrix using Expansion by Minors

Matrix:
$A = \begin{pmatrix} 1 & 2 & 0 & 3 \\ 4 & 5 & 6 & 7 \\ 0 & 8 & 9 & 1 \\ 2 & 1 & 3 & 4 \end{pmatrix}$

Solution:

Expanding along the first row:
$\text{det}(A) = 1 \cdot \text{det}\begin{pmatrix} 5 & 6 & 7 \\ 8 & 9 & 1 \\ 1 & 3 & 4 \end{pmatrix} – 2 \cdot \text{det}\begin{pmatrix} 4 & 6 & 7 \\ 0 & 9 & 1 \\ 2 & 3 & 4 \end{pmatrix} + 0 \cdot \text{det}\begin{pmatrix} 4 & 5 & 7 \\ 0 & 8 & 1 \\ 2 & 1 & 4 \end{pmatrix} + 3 \cdot \text{det}\begin{pmatrix} 4 & 5 & 6 \\ 0 & 8 & 9 \\ 2 & 1 & 3 \end{pmatrix}$

Calculate the 3×3 determinants:

$\text{det}\begin{pmatrix} 5 & 6 & 7 \\ 8 & 9 & 1 \\ 1 & 3 & 4 \end{pmatrix} = 5(9 \cdot 4 – 1 \cdot 3) – 6(8 \cdot 4 – 1 \cdot 1) + 7(8 \cdot 3 – 9 \cdot 1)$
$= 5(36 – 3) – 6(32 – 1) + 7(24 – 9)$
$= 5 \cdot 33 – 6 \cdot 31 + 7 \cdot 15$
$= 165 – 186 + 105$
$= 84$

$\text{det}\begin{pmatrix} 4 & 6 & 7 \\ 0 & 9 & 1 \\ 2 & 3 & 4 \end{pmatrix} = 4(9 \cdot 4 – 1 \cdot 3) – 6(0 \cdot 4 – 1 \cdot 2) + 7(0 \cdot 3 – 9 \cdot 2)$
$= 4(36 – 3) – 6(0 – 2) + 7(0 – 18)$
$= 4 \cdot 33 + 6 \cdot 2 – 7 \cdot 18$
$= 132 + 12 – 126$
$= 18$

$\text{det}\begin{pmatrix} 4 & 5 & 6 \\ 0 & 8 & 9 \\ 2 & 1 & 3 \end{pmatrix} = 4(8 \cdot 3 – 9 \cdot 1) – 5(0 \cdot 3 – 9 \cdot 2) + 6(0 \cdot 1 – 8 \cdot 2)$
$= 4(24 – 9) – 5(0 – 18) + 6(0 – 16)$
$= 4 \cdot 15 + 5 \cdot 18 – 6 \cdot 16$
$= 60 + 90 – 96$
$= 54$

Now, substitute back into the expansion:
$\text{det}(A) = 1 \cdot 84 – 2 \cdot 18 + 3 \cdot 54$
$= 84 – 36 + 162$
$= 210$

Answer: $$\text{det}(A) = 210$$.

#### Summary

The determinant is a scalar value that provides essential information about a square matrix, such as invertibility and properties of linear transformations. It is computed differently depending on the size of the matrix. Through the worked examples, we saw the determinants of 2×2, 3×3, and 4×4 matrices calculated using direct computation and expansion by minors.