Question

Determine the signs to be attached to the relevant minors in order to get the following cofactors of a determinant: $$\left|C_{13}|,| C_{231},\left|C_{33}\right|,\left|C_{41}\right|, \text { and }\left|C_{34} |\right.\right.$$

Short answer

Signs: +, +, +, -, -

Step-by-step solution

Understanding Cofactors

To find cofactors in a determinant, recall that cofactors are found by minor determinants. The sign of these cofactors is determined by the formula \[ (-1)^{i+j} \] where \( i \) is the row and \( j \) is the column.

Cofactor \( |C_{13}| \)

For the cofactor \( |C_{13}| \), we have \( i = 1 \) and \( j = 3 \). The sign is determined by \[ (-1)^{1+3} = (-1)^4 = +1 \].

Cofactor \( |C_{231}| \)

For the cofactor \( |C_{231}| \), which involves rows 2, 3, and 1, the last row is 1, and the last column is the third column in the minor formed. This makes \( i = 3 \) and \( j = 1 \). The sign is \[ (-1)^{3+1} = (-1)^4 = +1 \].

Cofactor \( |C_{33}| \)

For the cofactor \( |C_{33}| \), we have \( i = 3 \) and \( j = 3 \). The sign is determined by \[ (-1)^{3+3} = (-1)^6 = +1 \].

Cofactor \( |C_{41}| \)

For the cofactor \( |C_{41}| \), we have \( i = 4 \) and \( j = 1 \). The sign is determined by \[ (-1)^{4+1} = (-1)^5 = -1 \].

Cofactor \( |C_{34}| \)

For the cofactor \( |C_{34}| \), we have \( i = 3 \) and \( j = 4 \). The sign is determined by \[ (-1)^{3+4} = (-1)^7 = -1 \].

Key concepts

Determinants

The determinant is a special number that can be calculated from a square matrix. It is a useful quantity that provides information about the matrix, such as whether it is invertible or how it transforms space. In linear algebra, determinants are pivotal in helping to solve systems of linear equations, among other applications.To compute the determinant of a 2x2 matrix, you can use this simple formula:

• For matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \( ad - bc \).

When dealing with larger matrices, say 3x3 or 4x4, the calculation involves breaking down the matrix into smaller pieces using minors and cofactors, which we will describe in the following sections. A matrix with a determinant of zero is called singular, meaning it has no inverse, whereas a non-zero determinant implies it is invertible.

Cofactors

Cofactors play a crucial role in finding the determinant of larger matrices. Each cofactor originates from a minor, and it is modified by a sign that depends on the element's position in the matrix. The formula used to determine the sign of a cofactor is:\[(-1)^{i+j}\]where \(i\) is the row number and \(j\) is the column number of the element being considered.To calculate a cofactor, follow these steps:

• Identify the element of the matrix you are focusing on.
• Calculate the minor by removing the row and column of that element from the matrix.
• Apply the sign \((-1)^{i+j}\) to the minor.

The product obtained is the cofactor. Cofactors are used in the process of calculating the determinant of a matrix and are essential in computing the matrix's inverse.

Minors

A minor of a matrix is the determinant of a smaller sub-matrix, created by removing one row and one column from the original matrix. Minors are the building blocks for determining cofactors, and they help in expanding the determinant of larger matrices.Let's break down how to find a minor:

• Choose an element from the matrix, typically identified by its position \((i, j)\).
• Remove the \(i\)-th row and the \(j\)-th column from the matrix to form a smaller matrix.
• Calculate the determinant of this sub-matrix, which is your minor.

Minors are essential in matrix algebra, as they simplify complex matrices into manageable parts, enabling further calculations like cofactor expansion to find determinants of larger matrices.

Matrix Algebra

Matrix algebra is a collection of rules and operations involving matrices, which are rectangular arrays of numbers. This branch of algebra is widely used in various fields of science and engineering for solving systems of linear equations, transforming geometrical entities, and much more. Some key operations include:

• Matrix Addition/ Subtraction: Add or subtract matrices by adding or subtracting their corresponding elements.
• Scalar Multiplication: Multiply every entry of a matrix by a constant scalar.
• Matrix Multiplication: Multiply matrices together; this is more complex as it involves the dot product of rows and columns.
• Transpose of a Matrix: Switch the rows and columns of a matrix.

Understanding matrix algebra is fundamental. It provides the basis for more advanced concepts like linear transformations, eigenvalues, and eigenvectors that are essential in higher-level linear algebra.