Quantitative Aptitude - SPLessons

Determinants Formulas

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Determinants Formulas

shape Introduction

A factor which decisively affects the nature or outcome of something is called a Determinant. A quantity obtained by the addition of products of the elements of a square matrix according to a given rule.


shape Formulae

Matrices: A, B, C
Elements of a matrix: \(a_i, b_i, a_{ij}, b_{ij}, c_{ij}\)
Determinant of a matrix: det A
Minor of an element \( a_{ij}: M_{ij}\)
Cofactor of an element \( a_{ij}: C_{ij}\)
Transpose of a matrix: \( A^T, \widetilde{A}\)
Adjoint of a matrix: adj A
Trace of a matrix: tr A
Inverse of a matrix: \(A^-1\)
Real number: k
Real variables: \(x_i\)
Natural numbers: m, n


1. Second Order Determinant
det A = \(\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2 \\
\end{vmatrix}\) = \(a_1 b_2 -a_2 b_1\)


2. Third Order Determinant
det A = \(\begin{vmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}\\
\end{vmatrix}\) = \(a_{11} a_{22} a_{33} + a_{12} a_{23} a_ {31} + a_{13} a_{21} a_{31} –
a_{11} a_{23} a_{32} – a_{12} a_{21} a_{33} – a_{13} a_ {22} a_{31}\)


3. Third Order Determinant


4. N-th Order Determinant
det A = \(\begin{vmatrix}
a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2j} & \cdots & a_{2n}\\
\cdots & \cdots & \cdots & \cdots & \cdots & \\
a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in}\\
\cdots & \cdots & \cdots & \cdots & \cdots & \\
a_{n1} & a_{n2} & \cdots & a_{nj} & \cdots & a_{nn}\\
\end{vmatrix}\)


5. Minor
The minor \(M_{ij}\) associated with the element \(a_{ij}\) of n-th order matrix A is the (n -1)-th order determinant derived from the matrix A by deletion of its i-th row and j-th column.


6. Cofactor
\(C_{ij} = (-1)^{i+j} M_{ij}\)


7. Laplace Expansion of n-th Order Determinant
Laplace expansion by elements of the i-th row
det A = \(\displaystyle\sum_{j=1}^{n} a_{ij} C_{ij}, i = 1, 2…, n\)
Laplace expansion by elements of the j-th column
det A = \(\displaystyle\sum_{i=1}^{n}\)\(a_{ij} C_{ij}, i = 1, 2…, n\)



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