Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

If \(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\) are arbitrary square matrices of order 2 , show that \(|A B|=|A||B|\).

Step-by-Step Solution

Answer

The determinant of the matrix product AB equals the product of determinants: \(|AB| = |A||B|\).

Step 1: Definition of determinant for 2x2 matrices

For any 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is \(|A| = ad - bc\). Similarly for matrix B, \( B = \begin{pmatrix} e & f \ g & h \end{pmatrix} \), the determinant is \(|B| = eh - fg\).

Step 2: Definition of the matrix product AB

The product of two matrices \(A\) and \(B\) is \(AB = \begin{pmatrix} ae+bg & af+bh \ ce+dg & cf+dh \end{pmatrix}\).

Step 3: Calculate the determinant of AB

To find \(|AB|\), calculate the determinant:\[ (ae+bg)(cf+dh) - (af+bh)(ce+dg). \] Expand this expression using distributive property to get: \( acfh + aedh + bcgf + bdgh - afce - afdg - bhce - bhdg\).

Step 4: Simplify the deterministic expression

Simplify the expanded expression:\((acfh + aedh + bcgf + bdgh) - (afce + afdg + bhce + bhdg)\). After cancellation and reordering, you get:\[ (acfh - afce) + (aedh - afdg) + (bcgf - bhce) + (bdgh - bhdg) = (ad - bc)(eh - fg)\].

Step 5: Verify equality of determinants

The final expression can be rewritten as \(|A||B|\), confirming \(|AB| = |A||B|\) as \(|AB| = (ad - bc)(eh - fg) = |A||B|\).

Key Concepts

Determinant of 2x2 Matrix

Understanding the determinant of a 2x2 matrix is crucial for solving many matrix-related problems. A 2x2 matrix has the form \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]. The determinant of this matrix, represented as \(|A|\), tells us about the scaling factor of the transformation described by the matrix. Its computation is straightforward: \(|A| = ad - bc\).
This formula comes from observing how areas (or volumes in higher dimensions) change when geometric shapes are transformed by the matrix.
For instance, if \(|A|\) is zero, it implies that the matrix squashes any shape into a line or a point, leading to no area (in 2D).

• The determinant can indicate whether a matrix is invertible. If \(|A| = 0\), the matrix is not invertible; otherwise, it is.
• The sign of \(|A|\) can indicate orientation, with positive meaning no reversal in orientation.
• The absolute value of \(|A|\) gives the area scaling factor.

Matrix Multiplication

Matrix multiplication is a fundamental operation that combines two matrices, resulting in a new matrix. When multiplying two 2x2 matrices, each entry in the resulting matrix is computed by taking the dot product of rows from the first matrix and columns from the second matrix.
For example, if we have matrices \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\) and \(B = \begin{pmatrix} e & f \ g & h \end{pmatrix}\), the product \(AB\) is:\[ AB = \begin{pmatrix} ae+bg & af+bh \ ce+dg & cf+dh \end{pmatrix} \].
This operation is not commutative, meaning \(AB\) is generally not equal to \(BA\).

• The dimensions of the matrices must align for multiplication: the number of columns in A must equal the number of rows in B.
• Matrix multiplication is associative and distributive over addition.
• This operation is used in various applications, such as solving systems of linear equations and transformations in graphics.

Matrix Theory

Matrix theory is a broad area of mathematics involving the study of matrices and their properties. Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
In matrix theory, important concepts include matrix operations, the rank of a matrix, eigenvalues and eigenvectors, and special types of matrices like identity and diagonal matrices.
The study of matrix properties, such as determinants, assists in solving problems related to linear transformations and systems of linear equations. Here are some basic properties:

• An identity matrix acts like the number 1 in multiplication. Multiplying any matrix by the identity matrix leaves the original matrix unchanged.
• A diagonal matrix has non-zero elements only on its main diagonal. It simplifies multiplication significantly.
• Determinants help in understanding invertibility and the inherent changes a matrix imposes on space.

Matrix theory also extends into more complex topics like tensor products and block matrices, proving essential in advanced fields such as quantum mechanics, computer graphics, and machine learning.