Revision: Determinants and Matrices Maths HSC Science (General) 11th Standard Maharashtra State Board

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 matrix is a rectangular arrangement of numbers arranged in rows and columns, enclosed in brackets [ ] or parentheses ( ).

Elements (Entries) of a Matrix

• Each number in a matrix is called an element (or entry).

Rows and Columns

• Horizontal lines → rows
• Vertical lines → columns

Order of a Matrix

• Order = number of rows × number of columns
• Written as m × n and read as “m by n”

The transpose of a matrix is obtained by interchanging its rows and columns.

• If a matrix is A, its transpose is denoted by A^T

• If A is of order m × n, then
  A^T is of order n × m

• First row of A becomes first column of A^T, and so on.

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

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

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

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

• If D ≠ 0 → unique solution

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