If \(A = \(\left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right])\) , then trace (A 2) is equal to
2
Understanding the Matrix A and Hyperbolic Functions
The given matrix is \(A = \(\left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right])\). This matrix involves hyperbolic functions, specifically \(coshx\) and \(sinhx\). These functions have properties similar to trigonometric functions, and they appear often in various areas of mathematics and physics. The question asks for the Trace of Matrix A Squared, which means we first need to calculate the square of the matrix \(A\), denoted as \(A^2\), and then find its trace.
The trace of a square matrix is the sum of the elements on its main diagonal. For a 2x2 matrix \(\left[ {\begin{array}{} {a}&{b}\\ {c}&{d} \end{array}} \right]\), the trace is \(a+d\). To find the Trace of Matrix A Squared, we must perform matrix multiplication.
Calculating A Squared (\(A^2\)) using Matrix Multiplication
To find \(A^2\), we multiply matrix \(A\) by itself:
\(A^2 = A \times A = \left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right] \left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right]\)
Let's perform the matrix multiplication step-by-step:
- The element in the first row, first column of \(A^2\) is: \((coshx)(coshx) + (sinhx)(-sinhx) = cosh^2x - sinh^2x\)
- The element in the first row, second column of \(A^2\) is: \((coshx)(sinhx) + (sinhx)(coshx) = 2sinhx coshx\)
- The element in the second row, first column of \(A^2\) is: \((-sinhx)(coshx) + (coshx)(-sinhx) = -sinhx coshx - coshx sinhx = -2sinhx coshx\)
- The element in the second row, second column of \(A^2\) is: \((-sinhx)(sinhx) + (coshx)(coshx) = -sinh^2x + cosh^2x\)
So, the resulting matrix \(A^2\) is:
| \(cosh^2x - sinh^2x\) | \(2sinhx coshx\) |
| \(-2sinhx coshx\) | \(cosh^2x - sinh^2x\) |
Applying Hyperbolic Identities for the A Squared Calculation
Now, we can simplify the elements of \(A^2\) using fundamental identities of hyperbolic functions. The most relevant identity here is:
\(cosh^2x - sinh^2x = 1\)
Another useful identity is:
\(2sinhx coshx = sinh(2x)\)
Using these identities, we can simplify the elements of \(A^2\):
- The element in the first row, first column: \(cosh^2x - sinh^2x = 1\)
- The element in the first row, second column: \(2sinhx coshx = sinh(2x)\)
- The element in the second row, first column: \(-2sinhx coshx = -sinh(2x)\)
- The element in the second row, second column: \(cosh^2x - sinh^2x = 1\)
Therefore, the simplified matrix \(A^2\) is:
| \(1\) | \(sinh(2x)\) |
| \(-sinh(2x)\) | \(1\) |
Finding the Trace of Matrix A Squared
The final step is to find the trace of the simplified \(A^2\) matrix. The Trace of Matrix A Squared is the sum of the elements on its main diagonal.
\(trace(A^2) = (A^2)_{11} + (A^2)_{22}\)
From the simplified matrix \(A^2\), the diagonal elements are \(1\) and \(1\).
\(trace(A^2) = 1 + 1 = 2\)
So, the matrix trace of \(A^2\) is 2.
This result for the Trace of Matrix A Squared matches one of the given options.
Conclusion
After performing the A squared calculation and finding its trace using matrix multiplication and properties of hyperbolic functions like coshx sinhx, we found that the trace of \(A^2\) is 2.
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