91Ó°ÊÓ

In matrix algebra, calculating the determinant of a matrix is essential for understanding some of its fundamental properties. For a 2x2 matrix, like the matrix \( \mathbf{A} = \left[ \begin{array}{cc} 3 & 3 \ 2 & 4 \end{array} \right] \), the determinant can be calculated using a straightforward formula. The formula is:

• \( \text{det}(\mathbf{A}) = a \cdot d - b \cdot c \)

where \( a, b, c, \) and \( d \) are the elements of the matrix:

• \( a = 3, b = 3, c = 2, d = 4 \)

For our specific matrix, the calculation would look like this:

• \( \text{det}(\mathbf{A}) = 3 \times 4 - 3 \times 2 \)
• \( \text{det}(\mathbf{A}) = 12 - 6 \)
• \( \text{det}(\mathbf{A}) = 6 \)

The determinant tells us a lot about the matrix. For instance, if the determinant is zero, the matrix does not have an inverse. Since our determinant is 6, this matrix is invertible.

Finding the inverse of a matrix, particularly a 2x2 matrix, is a multi-step procedure that requires us to use more of what we know about matrix properties. Let’s revisit matrix \( \mathbf{A} = \left[ \begin{array}{cc} 3 & 3 \ 2 & 4 \end{array} \right] \). To find its inverse, we first need to calculate its determinant, which we already determined to be 6.
Here are the steps to find the inverse:

• Interchange the positions of \( a \) and \( d \), so you get a new matrix with \( 4 \) and \( 3 \) in the diagonal positions.
• Change the signs of \( b \) and \( c \). This means the off-diagonal elements \( 3 \) and \( 2 \) become \( -3 \) and \( -2 \).

This forms the adjugate of the matrix, which is:

• \( \begin{bmatrix} 4 & -3 \ -2 & 3 \end{bmatrix} \)

Finally, divide each element of the adjugate by the determinant (6):

• \( \mathbf{A}^{-1} = \frac{1}{6} \begin{bmatrix} 4 & -3 \ -2 & 3 \end{bmatrix} \)

With matrix algebra, ensuring the determinant isn't zero is crucial; otherwise, a matrix's inverse cannot be calculated.

Understanding the properties of a 2x2 matrix is fundamental when tackling such problems in matrix algebra. Here, we've dealt with a 2x2 matrix, \( \mathbf{A} = \left[ \begin{array}{cc} 3 & 3 \ 2 & 4 \end{array} \right] \), which beautifully illustrates the simplicity yet powerful nature of 2x2 matrices.
A 2x2 matrix has:

• The determinant as a key property that decides invertibility.
• Their elements can be easily interchanged or manipulated to find inverses or transposes.
• They are foundational blocks in linear algebra allowing simpler calculations before moving on to larger matrices.

Another property is that for 2x2 matrices, if the determinant is zero, it means the matrix is singular and thus does not have an inverse—highlighting again the determinant's centrality. The matrix \( \mathbf{A} \), with a non-zero determinant of 6, efficiently illustrates invertible property, thus making it a prime example to showcase these simple yet essential concepts within matrix algebra.