Question

Detine det \(\mathrm{A}\) and tind the determinant ot the tollowing matrices: (a) \(\left[a_{11}\right]\) (b) \(\begin{array}{ll}\mid a_{11} & a_{12} \mid \\ \mid a_{21} & a_{22} \mid\end{array}\) (c) \(\begin{array}{ccc} & 10 & 0 & 0 \\ & \mid 0 & 0 & 0 \\ & 0 & 0 & 0\end{array} \mid\) (d) \(\begin{array}{lll}\mid a_{11} & a_{12} & a_{13} \mid \\ \mid a_{21} & a_{22} & a_{23} \mid \\ \mid a_{31} & a_{32} & a_{33} \mid\end{array}\)

Short answer

In summary, the determinants for the given matrices are as follows: (a) \(\det(A) = a_{11}\) (b) \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\) (c) \(\det(A) = 0\) (d) \(\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\)

Step-by-step solution

Case (a)

Given a 1x1 matrix, the determinant is simply the value inside the matrix. For a matrix A with a single element, a11, we have: \[\det(A) = a_{11}\]

Case (b)

For a 2x2 matrix, we can find the determinant using the following formula: \[\det(A) = a_{11}a_{22} - a_{12}a_{21}\] Now, apply the formula to the given matrix: \[\det(A) = a_{11}a_{22} - a_{12}a_{21}\]

Case (c)

Given a 3x3 matrix with only the first element being non-zero (10) and others being 0, we can find the determinant using the rules of determinants: \[ \det(A) = \begin{vmatrix} 10 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{vmatrix} \] Since all the row elements except the top-left are zero, it simplifies to: \[\det(A) = 10 \cdot [(0 \cdot 0) - (0 \cdot 0)]\] \[\det(A) = 0\]

Case (d)

To find the determinant of a 3x3 matrix, we can follow this formula: \[ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \] Now, apply the formula to the given matrix: \[ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

Key concepts

Matrix Determinant

The determinant of a matrix is a special number that gives us a lot of information about the matrix itself. It is used in linear algebra to determine if a matrix is invertible, which means there is another matrix that can multiply it to produce the identity matrix. It also plays a role in finding the eigenvalues, which tell us about the matrix's various properties, such as its stability and much more.

For a 1x1 matrix, the determinant is simply the value of that one entry. As matrices grow larger, the process of finding a determinant gets more complex, involving various rules and operations including addition, multiplication, and subtraction. A non-zero determinant indicates that the matrix is invertible, while a zero determinant means it is singular, hence not invertible.

2x2 Matrix

A 2x2 matrix is a square array of numbers with two rows and two columns. The determinant of a 2x2 matrix is particularly straightforward to calculate. It’s found by crossing multiplying the diagonal entries and then subtracting one product from the other. The formula for the determinant of a matrix A, with entries \(a_{11}, a_{12}, a_{21}, a_{22}\), is:

\[\det(A) = a_{11}a_{22} - a_{12}a_{21}\]

This simple rule makes 2x2 matrices a nice introduction to the concept of determinants and their calculation.

3x3 Matrix

A 3x3 matrix expands on the complexity of 2x2 matrices, consisting of three rows and three columns. To find the determinant of a 3x3 matrix, there are several methods such as the rule of Sarrus or cofactor expansion. The cofactor expansion is often used for larger matrices and involves selecting a row or column, then calculating the determinant of smaller 2x2 matrices formed by eliminating the selected row and column. The general formula involves more multiplication and addition than the 2x2 case, accounting for three pairs of products, some of which are subtracted:

\[\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Understanding this process is essential because finding the determinant of larger matrices often involves breaking them down into smaller 3x3 cases.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vectors, vector spaces, linear mappings, and systems of linear equations. It provides the language and framework for many areas such as physics, computer science, engineering, and economics. Determinants play a key role in linear algebra because they are used in the calculation of eigenvalues, matrix inverses, and in understanding geometric properties of transformations. Knowing how to compute a matrix determinant is fundamental in linear algebra as it opens doors to more advanced topics and applications such as solving linear systems and understanding linear transformations.