“Rows then columns, always”
Reading the order of a matrix
A 2 × 3 matrix has 2 rows and 3 columns. The same order applies to the element aij, where i is the row.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Matrices and Determinants formulas
Matrices are mechanical marks: the rules are short, the arithmetic is the only risk, and solving a system of equations by matrices appears in almost every paper. Determinants supply the tests that decide whether a solution exists at all.
19 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Matrix basics
| What | Formula | Symbols and notes |
|---|---|---|
| Order | a matrix with m rows and n columns is m × n | rows first, always |
| Addition | possible only for the same order | add corresponding elements |
| Multiplication | possible when columns of the first equal rows of the second | the result is m × p |
| Not commutative | AB ≠ BA in general | even when both products exist |
| Transpose | (AB)T = BTAT | the order reverses |
| Symmetric matrix | AT = A | skew-symmetric when AT = −A, with zero diagonal |
Determinants
| What | Formula | Symbols and notes |
|---|---|---|
| Two by two | |A| = ad − bc | for rows (a b) and (c d) |
| Property, row swap | swapping two rows changes the sign | two identical rows give zero |
| Property, scalar | multiplying one row by k multiplies |A| by k | so |kA| = kn|A| for an n × n matrix |
| Product rule | |AB| = |A||B| | determinants multiply |
| Singular matrix | |A| = 0 | no inverse exists |
| Area of a triangle | ½|determinant of the coordinate matrix| | three collinear points give zero |
Inverse and systems
| What | Formula | Symbols and notes |
|---|---|---|
| Adjoint | adj A is the transpose of the cofactor matrix | cofactor Cij = (−1)i+jMij |
| Inverse | A−1 = adj A/|A| | exists only when |A| ≠ 0 |
| Key identity | A(adj A) = |A|I | the fastest route in many proofs |
| Determinant of adjoint | |adj A| = |A|n−1 | for an n × n matrix |
| Matrix method | X = A−1B | for the system AX = B |
| Consistency | |A| ≠ 0 gives a unique solution | |A| = 0 gives either no solution or infinitely many |
| Cramer's rule | x = D₁/D, y = D₂/D, z = D₃/D | D is the determinant of the coefficients |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
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Memory hooks
How to remember this chapter.
“Inner numbers must match”
When multiplication is possible
For an m × n times an n × p, the two inner numbers agree and the answer takes the outer ones: m × p.
“Zero determinant, no inverse”
Singular matrices
A−1 divides by |A|, so a zero determinant means no inverse and no unique solution to the system.
“Checkerboard signs”
Cofactors
Signs alternate plus, minus, plus across the matrix starting from the top-left. The sign is (−1)i+j.
Where marks go
The mistakes that cost the most.
- Assuming AB equals BA. Matrix multiplication is not commutative.
- Taking the transpose of a product without reversing the order.
- Dividing by a matrix. There is no division; multiply by the inverse instead.
- Forgetting that |kA| = kn|A|, not k|A|.
- Concluding no solution whenever |A| = 0. It may also mean infinitely many, and the two must be separated.
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