Problem 37 - Calculus for Biology and Medicine

Question

Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ 0 & -2 \end{array}\right] \text { and } I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ Show that \(A I_{2}=I_{2} A=A\).

Step-by-Step Solution

Answer

Multiplying \(A\) by the identity matrix \(I_2\) yields \(A\); hence, \(A I_2 = A\) and \(I_2 A = A\).

Step 1: Understand the Problem Statement

We need to verify that multiplying the matrix \(A\) by the identity matrix \(I_2\) yields \(A\) itself. In mathematical terms, we need to prove that \(A I_2 = A\) and \(I_2 A = A\).

Step 2: Identify the Matrices

The matrix \(A\) is given as \(\begin{bmatrix} 1 & 3 \ 0 & -2 \end{bmatrix}\) and the identity matrix \(I_2\) as \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). The identity matrix has the property that multiplying any matrix by it (of compatible size) results in the original matrix.

Step 3: Calculate \(AI_{2}\)

Let's perform the multiplication of matrix \(A\) by the identity matrix \(I_2\): \[A I_2 = \begin{bmatrix} 1 & 3 \ 0 & -2 \end{bmatrix} \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\] Multiply row by column:- First row: \((1)(1) + (3)(0) = 1\) and \((1)(0) + (3)(1) = 3\)- Second row: \((0)(1) + (-2)(0) = 0\) and \((0)(0) + (-2)(1) = -2\)Thus, \(A I_2 = \begin{bmatrix} 1 & 3 \ 0 & -2 \end{bmatrix}\), which is the same as \(A\).

Step 4: Calculate \(I_{2}A\)

Now perform the multiplication of the identity matrix \(I_2\) by the matrix \(A\): \[I_2 A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \ 0 & -2 \end{bmatrix}\]Multiply row by column:- First row: \((1)(1) + (0)(0) = 1\) and \((1)(3) + (0)(-2) = 3\)- Second row: \((0)(1) + (1)(0) = 0\) and \((0)(3) + (1)(-2) = -2\)Thus, \(I_2 A = \begin{bmatrix} 1 & 3 \ 0 & -2 \end{bmatrix}\), which is the same as \(A\).

Key Concepts

Identity Matrix

An identity matrix is a special kind of square matrix. Square matrices have the same number of rows and columns. For an identity matrix, all of the diagonal elements (the ones stretching from the upper left to the lower right) are 1s. All other elements are 0s. This creates a matrix that looks like this in its general form for a 2x2: \[I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\] When an identity matrix is multiplied by another matrix of the same size, it acts like the number 1 in regular multiplication. It does not change the other matrix. This is because the 1s on the diagonal "turn on" only the corresponding elements of the matrix they multiply, while the 0s "turn off" or cancel out everything else. Because of this, multiplying any 2x2 matrix by \(I_2\) will return the same 2x2 matrix. This property is very useful in matrix algebra, making identity matrices just as fundamental as the number 1 is in arithmetic.

2x2 Matrix

A 2x2 matrix consists of 2 rows and 2 columns, making it the simplest form of a square matrix. It’s a great starting point to understand larger matrices. Consider this matrix example: \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\] Here, each element in the matrix has its specific position, which is vital for operations such as matrix addition, subtraction, and multiplication. Let's take another look at matrix multiplication. When multiplying two 2x2 matrices, the elements of the resulting matrix are the sum of products across the rows of the first matrix and the columns of the second matrix. Here's a short breakdown of formula application:

• The first row, first column of the result: \((a)(1) + (b)(0)\)
• The first row, second column: \((a)(0) + (b)(1)\)
• The second row, first column: \((c)(1) + (d)(0)\)
• The second row, second column: \((c)(0) + (d)(1)\)

By practicing with 2x2 matrices like these, you build the skills needed to work with any larger matrices, enhancing your understanding of matrix algebra.

Matrix Algebra

Matrix algebra involves various operations that can be performed using matrices, such as addition, subtraction, scalar multiplication, and the focus here: matrix multiplication. Matrices are commonly used to solve systems of equations, optimize problems, and even in computer graphics. Matrix multiplication is a bit trickier than simple multiplication of numbers, as it requires specific conditions - like the number of columns in the first matrix must equal the number of rows in the second. If \(A\) is an \(m\times n\) matrix and \(B\) is an \(n\times p\) matrix, then their product \(C = AB\) is an \(m\times p\) matrix. Each element in the resulting matrix is calculated by this method: each element of the current row in the first matrix is multiplied by the corresponding element of the current column in the second matrix, and all these products are summed. It’s essential to practice with small matrices like 2x2 to fully grasp multiplication mechanics. This intuitive understanding of smaller matrices aids in managing more advanced matrix operations found in higher-level mathematics.