Chapter 4 — Determinants
One number, worked out from the entries of a square matrix, tells you whether a system of equations has a unique solution, how much a matrix stretches area, and whether the matrix can be inverted. That number is the determinant.
A surveyor in a Rajasthan village marks the three corners of a triangular field with a GPS receiver and gets three pairs of coordinates. A shopkeeper knows what three customers paid for different mixtures of rice, dal and sugar and wants the price of each item. A traffic engineer balancing the flow of cars through three junctions writes three equations in three unknowns. Each of them ends up with a square array of numbers — and in each case one particular number made from that array answers the first question that matters: how big is the field? is there exactly one set of prices? can the equations be solved at all?
That number is the determinant. In this chapter you will learn to compute it for matrices of order 1, 2 and 3 by expanding along any row or column, use it to find the area of a triangle and to test whether three points lie on a line, build the minors, cofactors and the adjoint of a matrix, prove the key identity $A(\operatorname{adj}A)=|A|\,I$, and from it get the formula for the inverse $A^{-1}=\tfrac1{|A|}\operatorname{adj}A$. Finally you will write a system of linear equations as $AX=B$, decide whether it is consistent, and solve it by $X=A^{-1}B$. Chapter 3 gave you matrices as objects to add and multiply; this chapter gives each square matrix a number that tells you what it can do.
- evaluate determinants of order 1, 2 and 3, expanding a $3\times3$ determinant along whichever row or column makes the work easiest, and solve equations such as $\begin{vmatrix}x&2\\8&x\end{vmatrix}=0$;
- find the area of a triangle from the coordinates of its vertices, test three points for collinearity, and write the equation of a line through two points as a determinant;
- write down the minor $M_{ij}$ and the cofactor $A_{ij}$ of any entry, and use cofactors to evaluate a determinant;
- build the adjoint of a $2\times2$ or $3\times3$ matrix, verify $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|\,I$, decide whether a matrix is singular, and find $A^{-1}$ when it exists;
- write a system of up to three linear equations as $AX=B$, decide whether it is consistent, and solve it by $X=A^{-1}B$ — including word problems on prices, mixtures and allocations;
- answer Board 1-, 2-, 3- and 5-mark questions and the JEE-style MCQs on determinants.
Sections in this chapter
Determinants and the Area of a Triangle
Determinants of order 1, 2 and 3; expansion along any row or column; area of a triangle, collinearity and the equation of a line.
Minors, Cofactors, Adjoint and Inverse
Minors and cofactors; the adjoint; $A(\operatorname{adj}A)=|A|\,I$; singular and non-singular matrices; the inverse by the adjoint formula.
Solving Systems of Linear Equations
Writing a system as $AX=B$; consistent and inconsistent systems; the matrix method $X=A^{-1}B$; what happens when $|A|=0$.