Grade 12th · Mathematics · Matrices and Determinants
Complete Class 12 formula cheat sheet for Matrices and Determinants (NCERT Chapters 3 & 4). Transpose properties, symmetric/skew-symmetric matrices, adjoint formulas, inverse matrix A⁻¹ = adj(A)/|A|, and Cramer's matrix method.
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Two matrices A (order m × n) and B (order p × q) can be multiplied only if n = p (columns of A = rows of B). The resulting matrix AB has order m × q. Key property: Matrix multiplication is associative [A(BC) = (AB)C] and distributive [A(B + C) = AB + AC], but generally NON-COMMUTATIVE: AB ≠ BA. Furthermore, AB = 0 does NOT imply that A = 0 or B = 0.
The adjoint of a square matrix A, denoted adj(A), is the transpose of the matrix of cofactors: adj(A) = [C_ij]^T. Fundamental identities for order n: 1. A · adj(A) = |A|·I. 2. |adj(A)| = |A|^(n - 1). 3. adj(AB) = adj(B) · adj(A). 4. |adj(adj(A))| = |A|^((n - 1)²). 5. adj(A^T) = (adj(A))^T.
A square matrix A is invertible (non-singular) if and only if |A| ≠ 0. If |A| = 0, A is singular and A⁻¹ does not exist. To solve a system of linear equations AX = B: 1. If |A| ≠ 0, the system has a unique consistent solution given by X = A⁻¹B = (1/|A|)·adj(A)·B. 2. If |A| = 0 and (adj(A))·B ≠ 0, the system is inconsistent and has NO solution. 3. If |A| = 0 and (adj(A))·B = 0, the system may be consistent with infinitely many solutions or inconsistent.
1. If any two rows (or columns) of a determinant are identical or proportional, |A| = 0. 2. If elements of a row/col are multiplied by k, value is k|A|. 3. |kA| = kⁿ|A| for order n. 4. |A^T| = |A|. 5. |AB| = |A|·|B|.
Always state the governing principle or formula first, write given values with standard SI units, substitute clearly, and highlight the final numerical answer with units in a box for full step marking.
Watch for unit prefixes (cm to m, minutes to seconds, g to kg, degrees to radians). In coordinate and ray optics problems, double-check Cartesian sign conventions before computing.
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The formula for the inverse of a square matrix A is A⁻¹ = (1 / |A|) × adj(A), provided the matrix is non-singular (|A| ≠ 0).
Using the formula |adj(A)| = |A|ⁿ⁻¹, for n = 3 and |A| = 5: |adj(A)| = 5³⁻¹ = 5² = 25.
For skew-symmetric matrix A of order n: A^T = -A. Taking determinants: |A^T| = |-A| = (-1)ⁿ|A|. Since |A^T| = |A|, we have |A| = (-1)ⁿ|A|. If n is odd, (-1)ⁿ = -1, so |A| = -|A| ⇒ 2|A| = 0 ⇒ |A| = 0.