<aside> 💡 Every square matrix $A$ has an associated number, called its determinant and is denoted by $det(A)$ or $|A|$.

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Determinants are used in:

• Solving a system of linear equations
• Finding inverse of a matrix
• Calculus and more.

• Determinant of a $1\times 1$ matrix: If $A = [a]$, a $1\times1$ matrix, then $det(A)=a$

• Determinant of $2\times2$ matrix:

  $A= \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $det(A) =ad-bc$

• Determinant of a $3\times 3$ matrix: $A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{bmatrix}$

  $det(A)= a_{11}\times det \begin{bmatrix} a_{22} & a_{23} \\a_{32} & a_{33} \end{bmatrix} - a_{12} \times det \begin{bmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{bmatrix} +a_{13} \times det \begin{bmatrix} a_{21} & a_{22} \\a_{31} & a_{32}\end{bmatrix}$

Determinant of Identity Matrix $I$

$det(I) = 1$

Determinant of a Product of Matrices

$det(AB)=det(A).det(B)$

Determinant of the Inverse of a Matrix

<aside> 💡 Inverse of matrix $A$ is denoted by $A^{-1}$ which satisfy: $AA^{-1} = I=A^{-1}A$

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$det(A^{-1})=\frac 1 {det(A)}$