Question

If \(P=\left[\begin{array}{ccc}0 & 4 & -2 \\ x & 0 & -y \\ 2 & -8 & 0\end{array}\right]\) is a skew \(-\) symmetric matrix, then \(x-y=\) ____. (1) 8 (2) 4 (3) \(-12\) (4) \(-8\)

Short answer

Answer: The value of x-y for the given skew-symmetric matrix P is 4.

Step-by-step solution

Write the Transpose of Matrix P

First, we will find the transpose of the given skew-symmetric matrix P by interchanging the rows and columns. The transpose of matrix P is written as: \(P^T = \left[\begin{array}{ccc} 0 & x & 2 \\ 4 & 0 & -8 \\ -2 & -y & 0 \end{array}\right]\)

Apply the skew-symmetric property

Now, we can apply the property of skew-symmetric matrices, which states that for the skew-symmetric matrix P, we have \(P^T = -P\). Thus, we can write the following equation, comparing the corresponding entries of both matrices: \(\left[\begin{array}{ccc} 0 & x & 2 \\ 4 & 0 & -8 \\ -2 & -y & 0 \end{array}\right] = \left[\begin{array}{ccc}-0 & -4 & 2 \\ -x & -0 & y \\ -2 & 8 & -0 \end{array}\right]\)

Extract relevant entries and compare

Comparing the corresponding entries, we can draw the following equalities between the entries of the two matrices: 1. \(x=-4\) 2. \(2=2\) 3. \(4=-x\) 4. \(-8=y\) 5. \(-2=-2\) 6. \(-y=8\)

Solve for x and y

From the above equalities, we can find the values of x and y as follows: Equality 1: \(x=-4\) Equality 6: \(-y=8 \Rightarrow y=-8\)

Compute x-y

Now that we have the value of x and y, we can compute the value of \(x-y\): \(x-y= (-4)-(-8)= -4+8=4\) Therefore, the value of \(x-y = 4\). Hence, the correct answer is option (2).

Key concepts

Transpose of Matrix

The transpose of a matrix is a new matrix derived by swapping the rows with the columns of the original matrix. If the original matrix is denoted by \(P\), its transpose is denoted as \(P^T\). This operation essentially flips the matrix over its diagonal.

• The element at the first row, second column in the original matrix, for example, becomes the element at the second row, first column in the transpose matrix.
• Formally, if \(P = [a_{ij}]\), then \(P^T = [a_{ji}]\).

For a matrix that's part of the exercise, performing the operation involves writing the elements of each row into columns of its transpose. This step is crucial when working with skew-symmetric matrices, as this type of matrix has specific properties tied to its transpose.

Matrix Properties

Matrices have various properties that are useful for solving problems in linear algebra. A skew-symmetric matrix, for instance, is a square matrix that is equal to the negative of its transpose. That means for a skew-symmetric matrix \(P\), the property is \(P^T = -P\).

• This property also implies that the diagonal elements of a skew-symmetric matrix must be zeros, since each diagonal element must be the negative of itself.
• These properties make it straightforward to derive important relationships between elements of the matrix.

Understanding these properties is fundamental for identifying other structural features in matrices, allowing for quick comparisons and solutions when handling equations, as seen in the exercise solution.

Matrix Equality

Two matrices are said to be equal only if they have the same dimensions and their corresponding elements are equal. In terms of solving equations involving matrices, this is a powerful concept.

• For example, if \(P^T = -P\) for the skew-symmetric matrix in the exercise, then each corresponding element in the transposed matrix and negative matrix must match.
• This equality of matrices allows us to set up a system of equations based on the equality of individual elements: \(a_{ij} = -b_{ij}\).

Both conditions of dimension compatibility and entry-wise equality hold significant importance when distinguishing between different types of matrices and working through solution steps as guided by matrix properties.

Matrix Equation Solving

Solving matrix equations often involves utilizing matrix properties and equality concepts in tandem. The goal is to find unknown elements in the matrix.

• In skew-symmetric matrices, the feature \(P^T = -P\) helps us form equations by comparing each corresponding element of the transpose matrix and its negative.
• This allows us to derive values for unknown variables, as seen in the exercise where we solve for \(x\) and \(y\) using the equations \(x = -4\) and \(y = -8\).
• With these values, mathematical operations such as finding \(x-y\) become straightforward, leading to solutions with much efficiency.

Proficiency in matrix equations comes down to understanding these processes and being able to set up correct equality statements for effective problem solving.