Matrices Solved Examples (Class 12 Mathematics)
Matrices numericals cover matrix operations, determinants, and applications in solving systems of equations. These problems are essential for linear algebr
TL;DR: Matrices numericals cover matrix operations, determinants, and applications in solving systems of equations. These problems are essential for linear a…
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Matrices numericals cover matrix operations, determinants, and applications in solving systems of equations. These problems are essential for linear algebr
Matrices — Solved Numerical Examples (Step by Step)
Example 1: Add matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
Solution: A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].
Example 2: Multiply matrices A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]].
Solution: A × B: Element (1,1) = 1×2 + 2×1 = 4. Element (1,2) = 1×0 + 2×3 = 6. Element (2,1) = 3×2 + 4×1 = 10. Element (2,2) = 3×0 + 4×3 = 12. A × B = [[4, 6], [10, 12]].
Example 3: Find the determinant of matrix [[3, 2], [1, 4]].
Solution: For 2×2 matrix [[a, b], [c, d]], determinant = ad - bc. det = 3×4 - 2×1 = 12 - 2 = 10.
Example 4: Find the inverse of matrix [[2, 1], [5, 3]].
Solution: For 2×2 matrix [[a, b], [c, d]], inverse = (1/det) × [[d, -b], [-c, a]]. det = 2×3 - 1×5 = 6 - 5 = 1. Inverse = 1 × [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]].
Example 5: Find the transpose of matrix [[1, 2, 3], [4, 5, 6]].
Solution: Transpose swaps rows and columns. A^T = [[1, 4], [2, 5], [3, 6]].
Example 6: Find the determinant of [[1, 2, 3], [0, 4, 5], [0, 0, 6]].
Solution: This is an upper triangular matrix. Determinant of triangular matrix is product of diagonal elements. det = 1 × 4 × 6 = 24.
Tips
- Matrix addition/subtraction: add/subtract corresponding elements.
- Matrix multiplication: (A×B)[i,j] = sum of A[i,k] × B[k,j] for all k.
- Determinant of triangular matrices = product of diagonal elements.
Frequently Asked Questions
When is a matrix invertible?
A square matrix is invertible if and only if its determinant is non-zero. Singular matrices (det = 0) do not have inverses.
What is the order of a matrix?
The order of a matrix is its dimensions: rows × columns. A matrix with 3 rows and 2 columns has order 3×2.
More Mathematics Solved Examples
- Quadratic Equations
- Trigonometry
- Arithmetic Progressions
- Surface Areas and Volumes
- Coordinate Geometry
- Probability
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