Applications of Matrices and Determinants — Study Notes (Class 12 Maths)

This chapter applies matrices and determinants to the central problem of linear algebra at school level: solving a system of linear equations and deciding when a solution exists. The tools below build up to four solving methods — matrix inversion, Cramer's rule, Gaussian elimination, and the rank method.

1. Non-singular matrix and its inverse

A square matrix \(A\) is non-singular when \(|A|\neq 0\) and singular when \(|A| = 0\). Only a non-singular matrix has an inverse \(A^{-1}\), and that inverse is unique. It is built from the adjoint:

\[ A^{-1} = \frac{1}{|A|}\,\operatorname{adj}A,\qquad |A|\neq 0. \]

2. Cofactors and the adjoint

Delete the \(i\)-th row and \(j\)-th column of \(A\) and take the determinant of what remains — that is the minor \(M_{ij}\). The cofactor attaches a sign: \(C_{ij} = (-1)^{i+j}M_{ij}\). The adjoint is the transpose of the cofactor matrix. A defining property is

\[ A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I. \]

This single identity quietly powers many MCQs: it instantly gives \(A(\operatorname{adj}A)\) once you know \(|A|\), and it is the reason \(A^{-1} = \tfrac{1}{|A|}\operatorname{adj}A\).

3. Properties worth memorising

• \(|\operatorname{adj}A| = |A|^{\,n-1}\) for an \(n\times n\) matrix.
• \(\operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A\), and consequently \(|\operatorname{adj}(\operatorname{adj}A)| = |A|^{\,(n-1)^2}\).
• \(\operatorname{adj}(AB) = (\operatorname{adj}B)(\operatorname{adj}A)\) — order reverses, exactly like the inverse.
• \(\operatorname{adj}(\lambda A) = \lambda^{\,n-1}\operatorname{adj}A\)  (note the exponent is \(n-1\), not \(n\)).
• \((\operatorname{adj}A)^{T} = \operatorname{adj}(A^{T})\).
• \((AB)^{-1} = B^{-1}A^{-1}\), \((A^{T})^{-1} = (A^{-1})^{T}\), \((A^{-1})^{-1} = A\), and \(|A^{-1}| = \tfrac{1}{|A|}\).

4. Orthogonal matrices

A square matrix \(A\) is orthogonal if \(AA^{T} = A^{T}A = I\). Equivalently, \(A\) is non-singular and \(A^{-1} = A^{T}\). The rows (and columns) then form an orthonormal set, which is the quickest way to pin down an unknown entry of a rotation-type matrix.

5. Elementary transformations, row-echelon form, and rank

Three elementary row operations — swapping two rows, scaling a row by a non-zero constant, and adding a multiple of one row to another — never change the rank of a matrix. Applying them reduces a matrix to row-echelon form (zeros below each leading entry, and lower rows no shorter than upper ones).

The rank \(\rho(A)\) is the order of the largest non-vanishing minor; in practice it is simply the number of non-zero rows in a row-echelon form. For an \(n\times n\) matrix, \(A^{-1}\) exists if and only if \(\rho(A) = n\).

6. Four ways to solve \(AX = B\)

When \(A\) is square and non-singular, any of the first three methods works:

• Matrix inversion: \(X = A^{-1}B\).
• Cramer's rule: \(x_i = \dfrac{\Delta_i}{\Delta}\), where \(\Delta = |A|\neq 0\) and \(\Delta_i\) replaces the \(i\)-th column of \(A\) by the constants \(B\).
• Gaussian elimination: reduce the augmented matrix \([A\,|\,B]\) to row-echelon form, then back-substitute. This is the most general method — it still works when \(A\) is singular or rectangular, which the other two cannot handle.

7. Consistency — the Rouché–Capelli rule

Compare two ranks, \(\rho(A)\) and \(\rho([A\,|\,B])\), with the number of unknowns \(n\):

• \(\rho(A) = \rho([A|B]) = n\) → consistent, unique solution.
• \(\rho(A) = \rho([A|B]) < n\) → consistent, infinitely many solutions (an \((n-\rho)\)-parameter family).
• \(\rho(A) \neq \rho([A|B])\) → inconsistent, no solution.

8. Homogeneous systems \(AX = O\)

These always have at least the trivial solution \(X = O\). A non-trivial solution exists precisely when \(\rho(A) < n\); for a square system that is the same as \(|A| = 0\). This is exactly the condition you set up whenever a problem says “has a non-trivial solution” — force the coefficient determinant to zero.

Common exam traps

• Mixing up \(\operatorname{adj}(\lambda A) = \lambda^{\,n-1}\operatorname{adj}A\) with \(\lambda^{n}\).
• Writing \(\operatorname{adj}(AB) = (\operatorname{adj}A)(\operatorname{adj}B)\) instead of reversing the order.
• Reading \(|A|\) as a modulus — a determinant can be negative.
• Claiming a unique solution from \(\rho(A) = \rho([A|B])\) alone, without comparing to \(n\).