Revision: Mathematics >> Determinants CUET (UG) Determinants

A determinant is a number associated with a square matrix.

\[\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}=ad-bc\]

The value of the determinant is ad - bc.

The degree of a 2 × 2 determinant is 2.

Cramer’s Rule is a method to solve simultaneous linear equations using determinants.

• It can be applied only when the determinant D ≠ 0

• $a1x+b1y=c1$

A determinant is a single real number associated with a square matrix only.

• Denoted by det ⁡A or ∣A∣ or Δ 

A determinant is a single real number associated with a square matrix only.

• Denoted by det ⁡A or ∣A∣ or Δ 

Consistent Solution: A system is consistent if it has at least one solution.

Inconsistent Solution: A system is inconsistent if it has no solution.

Consistent Solution: A system is consistent if it has at least one solution.

Inconsistent Solution: A system is inconsistent if it has no solution.

Let A = [a_ij] be a square matrix of order n. Then, the minor M_ij of a_ij in A is the determinant obtained by deleting the i^th row and the j^th column in which element a_ij lies. It is denoted by M_ij​ of A.

Let A = [a_ij] be a square matrix of order n. Then, the cofactor C_ij (or A_ij) of a_ij in A is (−1)^i+j times M_ij, where M_ij is the minor of a_ij in A.

∴ C_ij = (−1)^i+j M_ij

The adjoint of A is defined as the transpose (i.e. interchange rows and columns) of the cofactor matrix, and it is denoted by adj (A).

If A and B are non-singular square matrices of the same order such that AB = BA = I (where I is the identity matrix of the same order as A and B), then A and B are called inverses of each other.

We write A⁻¹ = B and B⁻¹ = A.

i.e. AA⁻¹ = A⁻¹A = I.

• If |A| ≠ 0, then A⁻¹ exists.
• If the inverse of a square matrix exists, then it is unique. A matrix can not have more than one distinct inverse.

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:

Consistent Solution: A system is consistent if it has at least one solution.

Inconsistent Solution: A system is inconsistent if it has no solution.

\[D=
\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix}=a_1b_2-a_2b_1\]

\[D_x=
\begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2
\end{vmatrix}=c_1b_2-c_2b_1\]

\[D_y=
\begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix}=a_1c_2-a_2c_1\]

\[x=\frac{D_x}{D}\quad\mathrm{and}\quad y=\frac{D_y}{D}\]

• If D ≠ 0 → unique solution

• If D = 0 → Cramer’s rule is not applicable

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

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

For \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \]

\[ \boxed{\text{adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}} \]

Shortcut

For a \[2 \times 2\] matrix:

• Interchange the diagonal elements.
• Change the signs of the off-diagonal elements.

• Minor \[M_{ij}\]: determinant of the matrix obtained by deleting row i and column j.

• Cofactor \[C_{ij}\]: \[C_{ij} = (-1)^{i+j}M_{ij}\].

• Determinant expansion along row i: \[|A| = \sum_{j=1}^{n} a_{ij}C_{ij}\].

• Determinant expansion along column j: \[|A| = \sum_{i=1}^{n} a_{ij}C_{ij}\].

• Determinant value is the same for any choice of row or column for expansion.

• Mixed row/column property: \[\sum_{j=1}^{n} a_{ij}C_{kj} = 0\] for \[i \neq k\].

• Matrix multiplication is row-by-column, not term-wise.

• Product AB exists only if columns of A = rows of B.

• If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].

• In general, \[AB \neq BA\], and sometimes one product may not even be defined.

• Matrix multiplication is associative and distributive over addition.

• Identity matrix acts as a multiplicative identity: AI = IA = A.

• Zero matrix absorbs multiplication: AO = OA = O.