Revision: Determinants Maths HSC Commerce (English Medium) 11th Standard Maharashtra State Board

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.

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

• $a1x+b1y=c1$

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

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

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

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

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

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

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

• If D ≠ 0 → unique solution

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