Problem 1 - Algebra and Trigonometry

Question

(a) The matrix \(I=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]\) is called an _____ matrix. (b) If \(A\) is a \(2 \times 2\) matrix, then \(A \times I=\) ______ and \(I \times A=\) _____ (c) If \(A\) and \(B\) are \(2 \times 2\) matrices with \(A B=I,\) then \(B\) is the ______ of \(A\)

Step-by-Step Solution

Answer

(a) Identity, (b) A, A, (c) Inverse

Step 1: Identify the Type of Matrix

The matrix \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \) is a special type of matrix. In matrix terms, this is called an identity matrix, as it has 1s along the diagonal and 0s elsewhere.

Step 2: Multiply a 2x2 Matrix by the Identity Matrix

Given a general \(2 \times 2\) matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), calculate \( A \times I \). Here, \( A \times I = \begin{bmatrix} a & b \ c & d \end{bmatrix} \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} = A \). This shows that multiplying a matrix by the identity matrix results in the original matrix, thus \( A \times I = A \).

Step 3: Multiply the Identity Matrix by a 2x2 Matrix

Similarly, calculate \( I \times A \). Here, \( I \times A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} = A \). This confirms that multiplying the identity matrix by any matrix results in the original matrix, maintaining \( I \times A = A \).

Step 4: Define an Inverse Matrix

The problem states \( AB = I \) for matrices \(A\) and \(B\), where \( I \) is the identity matrix. Thus, \(B\) is defined as the inverse of \(A\), which is a matrix such that when it multiplies \(A\), it results in the identity matrix. Therefore, \(B\) is the inverse of \(A\).

Key Concepts

Inverse Matrix

An inverse matrix is a key concept in linear algebra. It's a special type of matrix that essentially "reverses" the effect of another matrix during multiplication. If you have a matrix \(A\) and its inverse \(B\), their product will be the identity matrix \(I\), expressed as \(AB = I\). This is similar to multiplying a number by its reciprocal to get one (e.g., \( rac{1}{2} imes 2 = 1 \)). For a matrix to have an inverse, it must be square (i.e., have the same number of rows and columns) and not have a determinant of zero.
Here's why the inverse matrix is important:

• It enables the solution of matrix equations. If \(AX = B\), where \(A\) and \(B\) are matrices, knowing the inverse of \(A\) lets us find \(X\) as \(X = A^{-1}B\).
• It helps in transformations and understanding systems of equations.
• Inverse matrices are used in various fields like engineering, physics, and computer graphics.

When dealing with inverses, always remember they're unique for each matrix and provide a powerful tool to simplify complex problems.

Matrix Multiplication

Matrix multiplication is a method of multiplying two matrices to produce a new matrix. The process is different from arithmetic multiplication. For two matrices to be multiplied, their dimensions must follow the "inner dimensions must match" rule. Specifically, if you have matrix \(A\) with dimensions \(m \times n\) and matrix \(B\) with dimensions \(n \times p\), you can find the product \(C = AB\) which will have dimensions \(m \times p\).
Let's break down how it works:

• Each element in the resulting matrix \(C\) is found by taking the dot product of the corresponding row from \(A\) and column from \(B\).
• The element at row \(i\), column \(j\) of \(C\) is calculated as follows: \(C_{ij} = A_{i1}B_{1j} + A_{i2}B_{2j} + \, ...\, + A_{in}B_{nj}\).

Understanding matrix multiplication is crucial, especially because it's not commutative, meaning \(AB eq BA\) in general. Through this operation, we can perform transformations, solve systems of linear equations, and even change coordinate systems in geometry and physics.

Identity Property of Matrices

The identity matrix is a special type of matrix that's fundamental in matrix operations. It functions similarly to the number 1 in multiplication; it doesn't alter other numbers when used in multiplication. For any matrix \(A\), multiplying it by the identity matrix results in the same matrix: \(A \times I = A\) and \(I \times A = A\). This property underscores why the identity matrix is so named - it essentially "identifies" the other matrix.
Here are some key features:

• An identity matrix has 1s along its main diagonal and 0s elsewhere.
• It's always a square matrix, meaning it has the same number of rows and columns.
• The size of the identity matrix used must match the dimensions of the matrix it's multiplying.

The identity property is critical in linear algebra due to its role in simplification and equivalence transformations. In practice, it helps confirm matrix properties and assists in operations like finding inverses and proving concepts.