Definitions [2]
A determinant is a single real number associated with a square matrix only.
- Denoted by det A or ∣A∣ or Δ
Let \[A = [a_{ij}]\] be an \[m \times n\] matrix and \[B = [b_{jk}]\] be an \[n \times p\] matrix.
Then the product C = AB is an \[m \times p\] matrix \[C = [c_{ik}]\], where each entry \[c_{ik}\] is given by:
Formulae [1]
Order 1 (1×1 matrix):
∣A∣ = a
Order 2 (2×2 matrix):
∣A∣ = ad − bc
Order 3 (3×3 matrix):
\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\]
\[|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})\]
- If |A| = 0
A matrix is called a Singular Matrix - If |A| ≠ 0
Matrix is called a Non-Singular Matrix
Key Points
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Matrix multiplication is row-by-column, not term-wise.
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Product AB exists only if columns of A = rows of B.
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If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].
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In general, \[AB \neq BA\], and sometimes one product may not even be defined.
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Matrix multiplication is associative and distributive over addition.
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Identity matrix acts as a multiplicative identity: AI = IA = A.
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Zero matrix absorbs multiplication: AO = OA = O.
