Problem 1 - Precalculus: Mathematics for Calculus

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 matrix type

The matrix \(I=\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\) is a special type of matrix where all the elements on the main diagonal are 1 and all other elements are 0. This is known as the identity matrix. An identity matrix, when multiplied with any compatible matrix, yields the original matrix.

Step 2: Explore multiplication with identity matrix

Any matrix, say \(A\) of size \(2 \times 2\), when multiplied on either side by the identity matrix \(I\), remains unchanged, i.e., \(A \times I = A\) and \(I \times A = A\). This property holds because the identity matrix functions as the multiplicative identity for matrix multiplication.

Step 3: Determine the inverse relationship

If matrices \(A\) and \(B\) are such that their product is the identity matrix, i.e., \(A \times B = I\), then \(B\) is the inverse of \(A\). The inverse of a matrix, when multiplied with the original matrix, results in the identity matrix. The notation for the inverse of \(A\) is usually \(A^{-1}\).

Key Concepts

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves the rows of the first matrix and the columns of the second matrix. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. This is a key condition for performing matrix multiplication.

To multiply two matrices, say matrix \(A\) and matrix \(B\), each element in the resultant matrix \(C\) is obtained by multiplying the elements of the rows of \(A\) with the elements of the columns of \(B\) and summing them up. Specifically, the element in the \(i^{th}\) row and \(j^{th}\) column of \(C\) is calculated as follows:

• Take the \(i^{th}\) row of matrix \(A\).
• Take the \(j^{th}\) column of matrix \(B\).
• Multiply corresponding elements and sum them up.

The matrix multiplication results in a new matrix that has dimensions determined by the rows of the first matrix and columns of the second matrix. This operation is not commutative, which means \(A \times B eq B \times A\), generally. However, there are exceptions, such as when one of the matrices is an identity matrix.

inverse of a matrix

The inverse of a matrix is a special matrix that when multiplied by the original matrix results in an identity matrix. Not every matrix has an inverse. A matrix must be square (same number of rows and columns) and have a non-zero determinant to possess an inverse.

For a 2x2 matrix \(A\), the inverse, denoted as \(A^{-1}\), can be calculated using the formula:
\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]\

Where \(a, b, c, \) and \(d\) are the elements of the original matrix \(A\) in the format:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]\

The term \(ad-bc\) is known as the determinant of the matrix \(A\). If the determinant is zero, \(A\) does not have an inverse.

In practical applications, finding the inverse of a matrix is useful for solving linear equations, as it allows one to isolate variables in complex systems.

2x2 matrices

The 2x2 matrices are typically the simplest non-trivial square matrices you will encounter. These matrices have two rows and two columns, allowing them to perform basic operations like addition, subtraction, multiplication, and finding inverses with relative ease compared to larger matrices.

Here is an example of a general 2x2 matrix:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

Operations with 2x2 matrices are fundamental in solving systems of linear equations, especially when you can find the inverse. This type of matrix is sufficiently simple to allow for calculations by hand while still demonstrating key concepts of matrix mathematics.

For instance, when multiplying two 2x2 matrices, you follow specific rules by taking rows from the first matrix and columns from the second, performing element-wise multiplication, and summing results. Notably, properties such as the determinant help determine if further actions like finding inverses are possible. Hence, working with 2x2 matrices serves as a foundational skill in matrix algebra useful in various applications, including physics and engineering.