17. Introduction to Determinant
Dated: 21-04-2025
Determinant is a special number associated with a square matrix.1
\[
A =
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\]
\[\det(A) = |ad - bc|\]
\[|A| = |ad - bc|\]
One of the methods of taking a determinant of a \(3 \times 3\) matrix1
Definition
For \(n \ge 2\), the determinant of a matrix1 \(A_{n \times n}\) is defined as
\[\det(A) = \sum_{j = 1}^{n}(-1)^{i + j}a_{ij} \det(B_{(n - 1) \times( n - 1))})\]
Where \(B_{(n - 1) \times (n - 1)}\) is the matrix1 after deleting the \(i^{th}\) row and \(j^{th}\) column and is called the minor \(M_{ij}\) of the entry1 \([a_{ij}]\).
Each term of the summation is called the co-factor of entry1 \([a_{ij}]\).
Theorem
\(\det(A)\) can be computed with above summation formula but the index can be either \(i\) or \(j\).
Triangular Matrix
A matrix1 \(M_{m \times n}\) is called a triangular matrix if entries1 either below or above the diagonal, are \(0\).
Types
Upper Triangular
\[
\begin{bmatrix}
1 & 4 & 2 \\
0 & 3 & 4 \\
0 & 0 & 1
\end{bmatrix}
\]
Lower Triangular
\[
\begin{bmatrix}
1 & 0 & 0 \\
2 & 8 & 0 \\
4 & 9 & 7
\end{bmatrix}
\]
Gauss’ Algorithm for Evaluation of Determinants
- Convert the
matrix1 into upper triangular form. Determinantis the product of entries on thediagonal.
References
Read more about notations and symbols.