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

• $a1x+b1y=c1$

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

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

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

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

• If D ≠ 0 → unique solution

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

Let, Δ = `|(x,sintheta,costheta),(-sintheta,-x,1),(costheta,1,x)|`

= x(−x² − 1) − sin θ(−x sin θ − cos θ) + cos θ(−sin θ + x cos θ)

= −x(x² + 1) + x sin² θ + sin θ cos θ − sin θ cos θ + x cos² θ

= −x(x² + 1) + x(sin² θ + cos² θ)

= −x(x² + 1) + x

= −x[x² + 1 − 1]

= −x³