Square Matrix Determinant and Trace Evaluator

Easy

In linear algebra, every square matrix possesses two fundamental scalar properties that reveal crucial information about its structure and behavior: the determinant and the trace. These quantities are indispensable in understanding linear transformations, solving systems of equations, and analyzing matrix invertibility. The Determinant The determinant is a scalar value uniquely associated with square matrices that encapsulates essential geometric and algebraic properties. For a transformation represented by a matrix, the absolute value of its determinant indicates the scaling factor applied to volumes (or areas in 2D), while its sign reveals whether the transformation preserves or reverses orientation. For a 2×2 matrix: $$\begin{pmatrix} a & b \ c & d \end{pmatrix}$$ The determinant is computed as: $$\det(A) = ad - bc$$ For larger matrices, the determinant can be computed using various techniques such as cofactor expansion, LU decomposition, or recursive application of the 2×2 formula through minor matrices. Key Properties of Determinants: A matrix is invertible if and only if its determinant is non-zero The determinant of a product equals the product of determinants: det(AB) = det(A) × det(B) Row or column swaps negate the determinant Scaling a row/column by k multiplies the determinant by k The Trace The trace is a simpler yet equally important scalar: it is simply the sum of all diagonal elements (elements where row index equals column index). For a matrix A, the trace is defined as: $$\text{tr}(A) = \sum_{i=1}^{n} A_{ii}$$ Key Properties of Trace: The trace equals the sum of all eigenvalues of the matrix Trace is invariant under cyclic permutations: tr(ABC) = tr(BCA) = tr(CAB) Trace is a linear operation: tr(A + B) = tr(A) + tr(B) Your Task: Implement a function that takes a square matrix as input and returns a tuple containing both the determinant (as a floating-point number) and the trace (as a numeric value, preserving fractional diagonal sums). You may use NumPy or other numerical libraries for efficient computation.

Example 1

Input

matrix = [[2, 3], [1, 4]]

Output

(5.0, 6)

Explanation

For this 2×2 matrix: **Determinant Calculation:** Using the formula det(A) = ad - bc: • a = 2, b = 3, c = 1, d = 4 • det(A) = (2 × 4) - (3 × 1) = 8 - 3 = 5.0 **Trace Calculation:** Sum of diagonal elements (positions [0,0] and [1,1]): • trace = 2 + 4 = 6 The function returns the tuple (5.0, 6).

Example 2

Input

matrix = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Output

(1.0, 3)

Explanation

This is the 3×3 identity matrix, denoted I₃. **Determinant Calculation:** The determinant of any identity matrix is always 1. This makes intuitive sense: the identity transformation preserves volumes and orientation perfectly. • det(I₃) = 1.0 **Trace Calculation:** The diagonal elements are all 1: • trace = 1 + 1 + 1 = 3 The function returns (1.0, 3). Note that for any n×n identity matrix, det(Iₙ) = 1 and tr(Iₙ) = n.

Example 3

Input

matrix = [[4, -2], [1, 3]]

Output

(14.0, 7)

Explanation

For this 2×2 matrix with mixed positive and negative entries: **Determinant Calculation:** Using det(A) = ad - bc: • a = 4, b = -2, c = 1, d = 3 • det(A) = (4 × 3) - ((-2) × 1) = 12 - (-2) = 12 + 2 = 14.0 **Trace Calculation:** Sum of diagonal elements: • trace = 4 + 3 = 7 The function returns (14.0, 7). Notice how the negative off-diagonal element affects the determinant calculation but not the trace.

Constraints

1 ≤ n ≤ 100 (matrix is n × n)
The input will always be a valid square matrix (equal number of rows and columns)
-10⁶ ≤ matrix[i][j] ≤ 10⁶ for all elements
The determinant should be returned as a floating-point value
Preserve fractional trace values; round determinant and trace to two decimal places
You may use NumPy or similar numerical libraries

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Square Matrix Determinant and Trace Evaluator

Easy

In linear algebra, every square matrix possesses two fundamental scalar properties that reveal crucial information about its structure and behavior: the determinant and the trace. These quantities are indispensable in understanding linear transformations, solving systems of equations, and analyzing matrix invertibility. The Determinant The determinant is a scalar value uniquely associated with square matrices that encapsulates essential geometric and algebraic properties. For a transformation represented by a matrix, the absolute value of its determinant indicates the scaling factor applied to volumes (or areas in 2D), while its sign reveals whether the transformation preserves or reverses orientation. For a 2×2 matrix: $$\begin{pmatrix} a & b \ c & d \end{pmatrix}$$ The determinant is computed as: $$\det(A) = ad - bc$$ For larger matrices, the determinant can be computed using various techniques such as cofactor expansion, LU decomposition, or recursive application of the 2×2 formula through minor matrices. Key Properties of Determinants: A matrix is invertible if and only if its determinant is non-zero The determinant of a product equals the product of determinants: det(AB) = det(A) × det(B) Row or column swaps negate the determinant Scaling a row/column by k multiplies the determinant by k The Trace The trace is a simpler yet equally important scalar: it is simply the sum of all diagonal elements (elements where row index equals column index). For a matrix A, the trace is defined as: $$\text{tr}(A) = \sum_{i=1}^{n} A_{ii}$$ Key Properties of Trace: The trace equals the sum of all eigenvalues of the matrix Trace is invariant under cyclic permutations: tr(ABC) = tr(BCA) = tr(CAB) Trace is a linear operation: tr(A + B) = tr(A) + tr(B) Your Task: Implement a function that takes a square matrix as input and returns a tuple containing both the determinant (as a floating-point number) and the trace (as a numeric value, preserving fractional diagonal sums). You may use NumPy or other numerical libraries for efficient computation.

Example 1

Input

matrix = [[2, 3], [1, 4]]

Output

(5.0, 6)

Explanation

Example 2

Input

matrix = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Output

(1.0, 3)

Explanation

Example 3

Input

matrix = [[4, -2], [1, 3]]

Output

(14.0, 7)

Explanation

Constraints

1 ≤ n ≤ 100 (matrix is n × n)
The input will always be a valid square matrix (equal number of rows and columns)
-10⁶ ≤ matrix[i][j] ≤ 10⁶ for all elements
The determinant should be returned as a floating-point value
Preserve fractional trace values; round determinant and trace to two decimal places
You may use NumPy or similar numerical libraries