Matrices and Determinants Solved Examples (Class 12 Maths)
Matrices and determinants are powerful tools for solving systems of linear equations and performing transformations. These examples demonstrate matrix oper
TL;DR: Matrices and determinants are powerful tools for solving systems of linear equations and performing transformations. These examples demonstrate matrix…
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Matrices and determinants are powerful tools for solving systems of linear equations and performing transformations. These examples demonstrate matrix oper
Matrices and Determinants — Solved Numerical Examples (Step by Step)
Example 1: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find A + B.
Solution: Adding matrices element-wise:
A + B = [[1+5, 2+6], [3+7, 4+8]]
= [[6, 8], [10, 12]]
Example 2: Find the determinant of A = [[2, 3], [4, 5]].
Solution: For a 2×2 matrix [[a, b], [c, d]], determinant = ad - bc
det(A) = (2)(5) - (3)(4)
= 10 - 12
= -2
Example 3: Find the determinant of B = [[1, 2, 3], [0, 4, 5], [0, 0, 6]].
Solution: This is an upper triangular matrix. For triangular matrices, the determinant is the product of diagonal elements:
det(B) = 1 × 4 × 6 = 24
Example 4: Find the inverse of A = [[1, 2], [3, 4]].
Solution: For a 2×2 matrix [[a, b], [c, d]], the inverse is (1/det) × [[d, -b], [-c, a]]
First, find det(A) = (1)(4) - (2)(3) = 4 - 6 = -2
A⁻¹ = (1/-2) × [[4, -2], [-3, 1]]
= [[-2, 1], [3/2, -1/2]]
Example 5: Solve the system: 2x + y = 5, 3x + 2y = 8 using matrix method.
Solution: The system in matrix form is AX = B:
[[2, 1], [3, 2]] × [[x], [y]] = [[5], [8]]
det(A) = (2)(2) - (1)(3) = 4 - 3 = 1 ≠ 0, so inverse exists
A⁻¹ = (1/1) × [[2, -1], [-3, 2]] = [[2, -1], [-3, 2]]
X = A⁻¹B = [[2, -1], [-3, 2]] × [[5], [8]]
= [[2(5) + (-1)(8)], [(-3)(5) + 2(8)]]
= [[10 - 8], [-15 + 16]]
= [[2], [1]]
Therefore: x = 2, y = 1
Example 6: If A = [[1, 0], [0, 1]] (identity matrix), find A¹⁰.
Solution: The identity matrix I has the property that I × I = I.
Therefore:
A¹⁰ = I × I × I × ... (10 times) = I
A¹⁰ = [[1, 0], [0, 1]]
Tips
- For matrix multiplication AB, the number of columns in A must equal the number of rows in B.
- The determinant is zero if and only if the matrix is singular (non-invertible).
- For a matrix to have an inverse, its determinant must be non-zero.
- The transpose of a matrix A (denoted A^T) is obtained by swapping rows and columns; (A^T)^T = A.
Frequently Asked Questions
Is matrix multiplication commutative (AB = BA)?
No, matrix multiplication is generally not commutative. Even when both AB and BA are defined, they usually give different results. For example, with specific 2×2 matrices, AB ≠ BA in most cases. Only special matrices like the identity matrix commute with other matrices.
What is the significance of determinant being zero?
When the determinant of a matrix is zero, the matrix is singular and does not have an inverse. In the context of systems of linear equations, a zero determinant means the system either has no solution or infinitely many solutions, not a unique solution.
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