Determinants of order 2 and 3, area of a triangle, minors and cofactors, the adjoint and inverse of a matrix, and solving linear systems by the matrix method. Our own step-by-step solutions to Exercises 4.1 to 4.5 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. \(\begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad - bc\). For order 3, expand along any row or column with signs \(+ - +\); choose the row or column with the most zeros. \(|kA| = k^n|A|\) for an \(n \times n\) matrix.
What it tests. Area \(= \dfrac12\left|\begin{vmatrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{vmatrix}\right|\). Zero area means the points are collinear, which also gives the equation of the line through two points. When an area is given, use both signs of the determinant.
What it tests. The minor \(M_{ij}\) is the determinant left after deleting row i and column j; the cofactor is \(A_{ij} = (-1)^{i+j}M_{ij}\). A determinant equals the sum of the elements of any one row (or column) times their own cofactors.
What it tests. adj A is the transpose of the cofactor matrix; \(A(\operatorname{adj} A) = (\operatorname{adj} A)A = |A|I\). A is invertible exactly when \(|A| \ne 0\), and then \(A^{-1} = \dfrac{1}{|A|}\operatorname{adj} A\). For \(2 \times 2\): swap the diagonal, change the signs of the other two. If a matrix satisfies a polynomial equation, multiply by \(A^{-1}\) to get the inverse from it.
What it tests. Write the system as \(AX = B\). If \(|A| \ne 0\) there is exactly one solution, \(X = A^{-1}B\) (consistent). If \(|A| = 0\), work out \((\operatorname{adj} A)B\): non-zero means no solution (inconsistent); zero means either infinitely many solutions or none, which must then be checked directly.
What it tests. Expand along the row or column that keeps the arithmetic simplest (the rationalised book does not use row and column operations on determinants);for inverses of products use \((AB)^{-1} = B^{-1}A^{-1}\); systems in \(\tfrac1x, \tfrac1y, \tfrac1z\) become linear after substituting \(u = \tfrac1x\) and so on.
Determinants has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Determinants · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.