Determinants

All Mathematical truths are relative and conditional. — C. P. Steinmetz

4.1 Introduction

For a system

$a_1x+b_1y=c_1,\qquad a_2x+b_2y=c_2$

a unique solution exists when $a_1b_2-a_2b_1\mathrel{\htmlClass{course-neq}{\char"2260}} 0$. That number is the determinant of

$A=\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}$

This chapter treats determinants up to order $3$, minors, cofactors, adjoint, inverse, and solving linear systems.

$|A|$ means determinant of $A$, not modulus. Only square matrices have determinants.

4.2 Determinant

Order $1$

If $A=[a]$, then $|A|=a$.

Order $2$

$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}=a_{11}a_{22}-a_{21}a_{12}$

Order $3$ (expansion along a row or column)

Expanding along $R_1$:

$|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})$

Expanding along any row or column gives the same value. Prefer a row/column with many zeros.

If $A=kB$ and both are $n\times n$, then $|A|=k^n|B|$.

4.3 Properties (used constantly)

• $|A'|=|A|$
• Interchanging two rows (or columns) changes the sign of $|A|$
• If two rows (or columns) are identical, $|A|=0$
• $k$ times a row (or column) multiplies the determinant by $k$
• Adding a multiple of one row to another does not change $|A|$

4.4 Minors, cofactors, adjoint, inverse

Minor $M_{ij}$: determinant after deleting row $i$ and column $j$.

Cofactor: $A_{ij}=(-1)^{i+j}M_{ij}$

Adjoint: $\operatorname{adj} A$ is the transpose of the cofactor matrix.

$A(\operatorname{adj} A)=(\operatorname{adj} A)A=|A|I$

$A$ is singular if $|A|=0$, non-singular if $|A|\mathrel{\htmlClass{course-neq}{\char"2260}} 0$.

$A$ has an inverse if and only if $A$ is non-singular, and

$A^{-1}=\frac{1}{|A|}\operatorname{adj} A$

4.5 System of linear equations

Write $AX=B$. Unique solution:

$X=A^{-1}B\qquad\text{when }|A|\mathrel{\htmlClass{course-neq}{\char"2260}} 0$

• $|A|\mathrel{\htmlClass{course-neq}{\char"2260}} 0$: unique solution
• $|A|=0$ and $(\operatorname{adj} A)B\mathrel{\htmlClass{course-neq}{\char"2260}} 0$: no solution (inconsistent)
• $|A|=0$ and $(\operatorname{adj} A)B=0$: may or may not be consistent

Chapter Summary

• $|A|$ is defined only for square matrices
• Expand along any row or column
• Inverse exists $\iff$ $|A|\mathrel{\htmlClass{course-neq}{\char"2260}} 0$
• $A^{-1}=\dfrac{1}{|A|}\operatorname{adj} A$
• Unique solution of $AX=B$ is $X=A^{-1}B$ when $|A|\mathrel{\htmlClass{course-neq}{\char"2260}} 0$

Exercises (NCERT)

• Exercise 4.1 — evaluating determinants
• Later exercises — properties, adjoint, inverse, linear systems
• Miscellaneous Exercise