Let A be a skew-symmetric matrix of order 3. What is the value of det(4A4) - det(3A3) + det(2A2) - det(A) + det(-I) where I is the identity matrix of order 3?
-1
Understanding the Determinant of a Skew-Symmetric Matrix
The question asks us to evaluate an expression involving determinants of a skew-symmetric matrix of order 3. Let's first understand what a skew-symmetric matrix is and its key properties related to determinants.
A matrix A is called skew-symmetric if its transpose is equal to its negative, i.e., \(A^T = -A\). The given matrix A is a skew-symmetric matrix of order 3.
A crucial property of skew-symmetric matrices is related to their determinant based on their order:
- If a skew-symmetric matrix has an odd order (like 3, 5, etc.), its determinant is always zero.
- If a skew-symmetric matrix has an even order (like 2, 4, etc.), its determinant is a perfect square.
Since A is a skew-symmetric matrix of order 3 (which is odd), its determinant is zero. Thus, \(det(A) = 0\).
Evaluating Each Term in the Expression
The expression we need to evaluate is \(det(4A^4) - det(3A^3) + det(2A^2) - det(A) + det(-I)\), where I is the identity matrix of order 3.
We will evaluate each term separately using properties of determinants, specifically \(det(kA) = k^n det(A)\) where k is a scalar and n is the order of the matrix (here, n=3), and \(det(A^m) = (det(A))^m\).
Term 1: \(det(4A^4)\)
Using the property \(det(kA) = k^n det(A)\):
\(det(4A^4) = 4^3 det(A^4)\)
Using the property \(det(A^m) = (det(A))^m\):
\(det(A^4) = (det(A))^4\)
Since \(det(A) = 0\), we have \(det(A^4) = (0)^4 = 0\).
So, \(det(4A^4) = 4^3 \cdot 0 = 64 \cdot 0 = 0\).
Term 2: \(det(3A^3)\)
Using the property \(det(kA) = k^n det(A)\):
\(det(3A^3) = 3^3 det(A^3)\)
Using the property \(det(A^m) = (det(A))^m\):
\(det(A^3) = (det(A))^3\)
Since \(det(A) = 0\), we have \(det(A^3) = (0)^3 = 0\).
So, \(det(3A^3) = 3^3 \cdot 0 = 27 \cdot 0 = 0\).
Term 3: \(det(2A^2)\)
Using the property \(det(kA) = k^n det(A)\):
\(det(2A^2) = 2^3 det(A^2)\)
Using the property \(det(A^m) = (det(A))^m\):
\(det(A^2) = (det(A))^2\)
Since \(det(A) = 0\), we have \(det(A^2) = (0)^2 = 0\).
So, \(det(2A^2) = 2^3 \cdot 0 = 8 \cdot 0 = 0\).
Term 4: \(det(A)\)
As established earlier, for a skew-symmetric matrix of order 3, \(det(A) = 0\).
Term 5: \(det(-I)\)
Here, I is the identity matrix of order 3. \(-I\) is the matrix obtained by multiplying the identity matrix by -1.
Using the property \(det(kA) = k^n det(A)\) with \(k=-1\) and \(A=I\), which is of order n=3:
\(det(-I) = det((-1)I) = (-1)^3 det(I)\)
The determinant of an identity matrix of any order is 1, i.e., \(det(I) = 1\).
So, \(det(-I) = (-1)^3 \cdot 1 = -1 \cdot 1 = -1\).
Calculating the Final Value of the Expression
Now, substitute the values of the individual terms back into the expression:
\(det(4A^4) - det(3A^3) + det(2A^2) - det(A) + det(-I)\)
\(= 0 - 0 + 0 - 0 + (-1)\)
\(= -1\)
The value of the expression is -1.
Revision Table: Skew-Symmetric Matrix Determinant
Here's a quick summary of the properties used:
| Concept | Property | Application in this problem (n=3) |
|---|---|---|
| Skew-symmetric matrix | \(A^T = -A\) | A is of order 3 and skew-symmetric |
| Determinant of Skew-symmetric (Odd Order) | If A is skew-symmetric of odd order n, \(det(A) = 0\) | \(det(A) = 0\) (since order is 3) |
| Determinant of Scalar Multiple | \(det(kA) = k^n det(A)\) | \(det(4A^4) = 4^3 det(A^4)\), \(det(3A^3) = 3^3 det(A^3)\), \(det(2A^2) = 2^3 det(A^2)\), \(det(-I) = (-1)^3 det(I)\) |
| Determinant of Matrix Power | \(det(A^m) = (det(A))^m\) | \(det(A^4) = (det(A))^4\), \(det(A^3) = (det(A))^3\), \(det(A^2) = (det(A))^2\) |
| Determinant of Identity Matrix | \(det(I) = 1\) | \(det(I) = 1\) (for order 3) |
Additional Information: Matrix and Determinant Properties
Understanding the properties of matrices and determinants is fundamental in linear algebra. Here are a few more points related to the concepts used in this problem:
- The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible (a matrix is invertible if and only if its determinant is non-zero).
- For any square matrices A and B of the same order, \(det(AB) = det(A) det(B)\). This property was implicitly used when we stated \(det(A^m) = (det(A))^m\), as \(A^m\) is a product of A with itself m times.
- The determinant of a transpose matrix is equal to the determinant of the original matrix, i.e., \(det(A^T) = det(A)\).
- Identity matrix (I) is a square matrix with ones on the main diagonal and zeros elsewhere. It acts like '1' in matrix multiplication, i.e., \(AI = IA = A\).
- Scalar multiplication of a matrix A by k results in a matrix where every element of A is multiplied by k. The determinant scales by \(k^n\), where n is the order.
These properties are essential for solving various problems involving matrices and determinants.
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