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 AA be an involutory matrix, meaning A2=IA^2 = I, where II is the identity matrix. For a 2×22 \times 2 matrix, we will calculate the value of 3(trace(A))2+4(det(A))23(\text{trace}(A))^2 + 4(\text{det}(A))^2.

Step 1: General form of a 2×22 \times 2 matrix

Let AA be a general 2×22 \times 2 matrix:

A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

For an involutory matrix, we know A2=IA^2 = I. Let's use this property.

Step 2: Condition for an involutory matrix

From A2=IA^2 = I, we expand A2A^2:

A2=(abcd)(abcd)=(a2+bcab+bdac+cdbc+d2)A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix}

Equating this to the identity matrix I=(1001)I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, we get the following system of equations:

  1. a2+bc=1a^2 + bc = 1
  2. ab+bd=0ab + bd = 0
  3. ac+cd=0ac + cd = 0
  4. bc+d2=1bc + d^2 = 1

Step 3: Trace and determinant of AA

The trace of matrix AA is:

trace(A)=a+d\text{trace}(A) = a + d

The determinant of matrix AA is:

det(A)=adbc\text{det}(A) = ad - bc

Step 4: Squaring trace and determinant

We are asked to find the value of:

3(trace(A))2+4(det(A))23(\text{trace}(A))^2 + 4(\text{det}(A))^2

Let t=trace(A)=a+dt = \text{trace}(A) = a + d and Δ=det(A)=adbc\Delta = \text{det}(A) = ad - bc. Then the expression becomes:

3t2+4Δ23t^2 + 4\Delta^2

Step 5: Using the involutory property

From the property A2=IA^2 = I, it can be shown that for a 2×22 \times 2 involutory matrix, the trace tt satisfies:

t2=4t^2 = 4

And the determinant of an involutory matrix is always:

Δ=1\Delta = 1

Step 6: Substituting values

Now substitute t2=4t^2 = 4 and Δ=1\Delta = 1 into the expression:

3(4)+4(1)2=12+4=163(4) + 4(1)^2 = 12 + 4 = 16

Final Answer:

The value of 3(trace(A))2+4(det(A))23(\text{trace}(A))^2 + 4(\text{det}(A))^2 is 16\boxed{16}.

Would you like more details or have any questions about the solution?

Here are 5 related questions to expand your understanding:

  1. What is an involutory matrix and how can we recognize it?
  2. Why is the determinant of an involutory matrix always 1?
  3. Can a non-diagonal matrix be involutory? Give an example.
  4. How does the trace of a matrix relate to its eigenvalues?
  5. 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