Math Problem Statement
For an involutory matrix A (elements of whose principal diagonal are not same) of order 2 × 2, the value of 3(trace (A))² + 4(det (A))² is equal to
Solution
Let matrix be an involutory matrix, meaning , where is the identity matrix. For a matrix, we will calculate the value of .
Step 1: General form of a matrix
Let be a general matrix:
For an involutory matrix, we know . Let's use this property.
Step 2: Condition for an involutory matrix
From , we expand :
Equating this to the identity matrix , we get the following system of equations:
Step 3: Trace and determinant of
The trace of matrix is:
The determinant of matrix is:
Step 4: Squaring trace and determinant
We are asked to find the value of:
Let and . Then the expression becomes:
Step 5: Using the involutory property
From the property , it can be shown that for a involutory matrix, the trace satisfies:
And the determinant of an involutory matrix is always:
Step 6: Substituting values
Now substitute and into the expression:
Final Answer:
The value of is .
Would you like more details or have any questions about the solution?
Here are 5 related questions to expand your understanding:
- What is an involutory matrix and how can we recognize it?
- Why is the determinant of an involutory matrix always 1?
- Can a non-diagonal matrix be involutory? Give an example.
- How does the trace of a matrix relate to its eigenvalues?
- Can we generalize this result to higher-dimensional involutory matrices?
Tip: When solving matrix equations, using the matrix's properties (like trace, determinant, and special conditions like involutory) can significantly simplify the computation.
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Trace of a Matrix
Determinant of a Matrix
Involutory Matrices
Formulas
trace(A) = a + d
det(A) = ad - bc
A^2 = I (involutory property)
3(trace(A))² + 4(det(A))²
Theorems
Involutory Matrix Theorem
Suitable Grade Level
Undergraduate