Question

Given \(\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right|\) find the minors and cofactors of the elements \(a, b,\) and \(f\)

Short answer

Minor for \( a \): \( ei - fh \), cofactor: \( ei - fh \). Minor for \( b \): \( di - fg \), cofactor: \( fg - di \). Minor for \( f \): \( ah - bg \), cofactor: \( bg - ah \).

Step-by-step solution

Identify Minor for Element a

To find the minor of element \( a \), eliminate the first row and first column from the determinant. You will be left with the following matrix: \[ \begin{vmatrix} e & f \ h & i \end{vmatrix} \]. The determinant of this matrix is calculated as \( ei - fh \). This is the minor of \( a \).

Calculate Cofactor for Element a

Use the formula for the cofactor, which is \( C_{ij} = (-1)^{i+j} \cdot M_{ij} \). For element \( a \), located at position (1,1), the cofactor is \( C_{11} = (-1)^{1+1} \cdot (ei - fh) = ei - fh \).

Identify Minor for Element b

For element \( b \), remove the first row and second column, resulting in the matrix \[ \begin{vmatrix} d & f \ g & i \end{vmatrix} \]. The determinant here is \( di - fg \). This is the minor of \( b \).

Calculate Cofactor for Element b

The cofactor for element \( b \), located at position (1,2), is \( C_{12} = (-1)^{1+2} \cdot (di - fg) = -(di - fg) = fg - di \).

Identify Minor for Element f

For element \( f \), eliminate the second row and third column, resulting in the matrix \[ \begin{vmatrix} a & b \ g & h \end{vmatrix} \]. The determinant is \( ah - bg \). This is the minor of \( f \).

Calculate Cofactor for Element f

The cofactor for element \( f \), located at position (2,3), is \( C_{23} = (-1)^{2+3} \cdot (ah - bg) = -(ah - bg) = bg - ah \).

Key concepts

Determinants

Determinants play a crucial role in matrix algebra, acting as a scalar value that gives insight into various matrix properties. When you assess determinants, think of them as sorting out whether a matrix can be inverted or not. For instance, a matrix with a determinant of zero cannot be inverted. This property is handy in solving systems of linear equations.
To calculate a determinant for a 3x3 matrix, we apply a specific formula. You choose an element from the first row and then multiply it by the determinant of the 2x2 matrix obtained by removing the first row and the column of the chosen element. You repeat this for all elements in the first row, alternating the signs.
Here's a simple illustration for a 3x3 matrix \[ \left|\begin{array}{lll}a & b & c \ d & e & f \ g & h & i\end{array}\right| \] The determinant would be calculated as:

• \[ a \times (ei - fh) \]
• \[ -b \times (di - fg) \]
• \[ +c \times (dh - eg) \]

By calculating these values and summing them, you arrive at the determinant, which helps understand the behavior of the matrix.

Minors and Cofactors

Minors and cofactors are essential when working with determinants and solving matrix-related problems. A minor of an element in a matrix is the determinant of a smaller matrix, obtained by removing the row and column of that particular element. For example, if you want the minor for element \( a \) in a 3x3 matrix, you eliminate its row and column, leaving you with a new 2x2 matrix.

Once you have the minor, determining the cofactor is the next step. A cofactor is derived by multiplying the minor by \( (-1)^{i+j} \), where \( i \) is the row number and \( j \) is the column number of the element in the original matrix. This step introduces alternating signs into the calculation, crucial for accurately expressing a determinant.

For instance, to find the cofactor of the element \( a \) at position \( (1,1) \), you would calculate it as:

• Find the minor: \( ei - fh \)
• Calculate the cofactor: \( C_{11} = (-1)^{1+1} \times (ei - fh) = ei - fh \)

Through these steps, minors and cofactors help simplify the process of finding a matrix determinant and are foundational concepts in matrix theory.

Matrix Algebra

Matrix algebra is a robust area of mathematics that gives us the tools to solve complex systems of equations, transform geometries, and perform various computational techniques. At its core, it involves mathematical operations specific to matrices, such as addition, subtraction, multiplication, and finding inverses.

Understanding that matrices can represent large sets of equations concurrently is one of the great strengths of matrix algebra. It simplifies problems that would otherwise be bulky if tackled using traditional algebraic methods. Operations like matrix multiplication might seem tricky at first, as it differs from multiplying single numbers. The key takeaway in matrix multiplication is that the order matters; \( AB \) is not the same as \( BA \), as matrices are not generally commutative.

Another interesting aspect is the role of identity and zero matrices in matrix algebra. The identity matrix operates similarly to the number 1 in scalar multiplication—it does not change the matrix it multiplies. On the other hand, the zero matrix, much like 0, nullifies any matrix it multiplies.

By mastering these concepts, you unlock a toolkit for exploring more advanced topics like eigenvectors and eigenvalues, ultimately enriching your understanding of linear transformations.