Understanding Skew-Symmetric Matrices

A matrix \( A \) is defined as skew-symmetric if its transpose \( A^T \) is equal to the negative of the matrix \( -A \). Mathematically, this property is expressed as \( A^T = -A \).

For a matrix \( A = [a_{ij}] \), being skew-symmetric means that for all elements, \( a_{ij} = -a_{ji} \). A direct consequence of this property is that the diagonal elements must be zero, because for \( i=j \), \( a_{ii} = -a_{ii} \), which implies \( 2a_{ii} = 0 \), so \( a_{ii} = 0 \).

Applying the Skew-Symmetric Property to the Given Matrix

The given matrix is:

Let's find the transpose of matrix \( A \), denoted by \( A^T \). The transpose is obtained by swapping rows and columns.

Now, let's find the negative of matrix \( A \), denoted by \( -A \). This is done by multiplying each element of \( A \) by -1.

Equating Matrices \( A^T \) and \( -A \)

For the matrix \( A \) to be skew-symmetric, we must have \( A^T = -A \). Let's equate the corresponding elements of the two matrices:

By equating the elements at corresponding positions, we get the following equations:

1. \( 0 = 0 \) (Diagonal element, consistent)
2. \( a = 3 \) (Element at (1, 2))
3. \( -2 = -2 \) (Element at (1, 3), consistent)
4. \( -3 = -a \) (Element at (2, 1))
5. \( b = -b \) (Diagonal element at (2, 2))
6. \( c = -8 \) (Element at (2, 3))
7. \( 2 = 2 \) (Element at (3, 1), consistent)
8. \( 8 = -c \) (Element at (3, 2))
9. \( 0 = 0 \) (Diagonal element at (3, 3), consistent)

Solving for a, b, and c

From the equations above, we can determine the values of a, b, and c:

• From equation 2 or 4: \( a = 3 \).
• From equation 5: \( b = -b \implies 2b = 0 \implies b = 0 \). This confirms that the diagonal element must be zero, as expected for a skew-symmetric matrix.
• From equation 6: \( c = -8 \).
• From equation 8: \( 8 = -c \implies c = -8 \). This is consistent with equation 6.

So, for the given matrix to be skew-symmetric, we must have \( a = 3 \), \( b = 0 \), and \( c = -8 \).

Checking the Options

Now let's examine the given options based on these values:

• Option 1: \( b = 0 \) and \( a + c = 5 \). We found \( b=0 \) which is correct. However, \( a + c = 3 + (-8) = -5 \), which is not equal to 5. So, this option is incorrect.
• Option 2: \( a = 0 \) and \( b + c = 5 \). We found \( a=3 \), not 0. So, this option is incorrect.
• Option 3: \( b = 0 \) and \( a + c = -5 \). We found \( b=0 \) which is correct. Also, \( a + c = 3 + (-8) = -5 \), which matches the condition \( a+c = -5 \). So, this option is correct.
• Option 4: \( c = 0 \) and \( a + b = -5 \). We found \( c=-8 \), not 0. So, this option is incorrect.

Based on our calculations, the conditions \( b = 0 \) and \( a + c = -5 \) must be true for the matrix to be skew-symmetric.

Revision Table: Skew-Symmetric Matrix Properties

Additional Information: Related Matrix Concepts

Understanding skew-symmetric matrices is part of learning about different types of special matrices in linear algebra. Here are a few related concepts:

• Symmetric Matrix: A matrix \( A \) is symmetric if \( A^T = A \). This means \( a_{ij} = a_{ji} \) for all i, j. Unlike skew-symmetric matrices, diagonal elements can be non-zero.
• Transpose of a Matrix: The transpose \( A^T \) of a matrix \( A \) is obtained by interchanging its rows and columns. If \( A \) is an \( m \times n \) matrix, \( A^T \) is an \( n \times m \) matrix.
• Matrix Addition and Scalar Multiplication: These are fundamental operations used when working with matrices, including properties like \( -(A) = (-1)A \).
• Square Matrix: Skew-symmetric matrices, like symmetric matrices, must be square matrices (number of rows equals number of columns). The given matrix is a \( 3 \times 3 \) matrix, which is a square matrix.

These concepts are important for solving problems involving matrix properties and transformations.