Problem 22 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Compute } A B C $$

Step-by-Step Solution

Answer

The matrix product \(ABC\) is \(\left[\begin{array}{cc} -2 & -1 \\ 0 & -5 \end{array}\right]\).

Step 1: Compute AB

To find the product of matrices \(A\) and \(B\), multiply the rows of \(A\) by the columns of \(B\). The resulting matrix will be: \[ AB = \left[\begin{array}{cc} (-1 \cdot 2 + 0 \cdot (-1)) & (-1 \cdot 3 + 0 \cdot 1) \ (1 \cdot 2 + 2 \cdot (-1)) & (1 \cdot 3 + 2 \cdot 1) \end{array}\right] \] Simplifying each entry, we get: \[ AB = \left[\begin{array}{cc} -2 & -3 \ 0 & 5 \end{array}\right] \]

Step 2: Compute (AB)C

Now, multiply the resulting matrix from Step 1, \(AB\), by matrix \(C\). We perform the multiplication similarly as before: \[ ABC = \left[\begin{array}{cc} (-2 \cdot 1 + (-3) \cdot 0) & (-2 \cdot 2 + (-3) \cdot (-1)) \ (0 \cdot 1 + 5 \cdot 0) & (0 \cdot 2 + 5 \cdot (-1)) \end{array}\right] \] Simplifying each entry, we have: \[ ABC = \left[\begin{array}{cc} -2 & -1 \ 0 & -5 \end{array}\right] \]

Key Concepts

Matrix Operations

Matrix operations involve various actions such as addition, subtraction, and multiplication, but this exercise focuses on multiplication. When multiplying matrices, it's crucial to check their dimensions first. The number of columns in the first matrix must match the number of rows in the second matrix to perform multiplication successfully.

Let's consider matrices \(A\) and \(B\) from the exercise. Matrix \(A\) has dimensions \(2 \times 2\) (2 rows and 2 columns), and matrix \(B\) also has dimensions \(2 \times 2\). Hence, they can be multiplied directly. The multiplied result will also be a \(2 \times 2\) matrix.

Matrix multiplication follows a row-by-column rule. Specifically, this means you multiply each element of a row in the first matrix by each element of a column in the second matrix, and then sum up these products to get a single entry in the resulting matrix. It may seem daunting at first, but it becomes straightforward with practice. Remember:

• The first element in the new matrix comes from multiplying corresponding elements of the first row of the first matrix by the first column of the second matrix.
• Continue this process for each combination, filling out the new matrix row by row.

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, matrix operations, and the study of linear transformations. It’s widely used in various fields, including engineering, computer science, and economics.

A fundamental part of linear algebra is understanding how matrices interact with each other through operations like addition and multiplication. Multiplying matrices, as seen here, is not only about calculating numbers but is also essential in performing transformations, handling data shifts, rotations, and reflections.

In our exercise, when we compute \(ABC\), each matrix multiplication step represents a transformation applied to either a space or data. Understanding the theory behind matrix multiplication helps you visualize and solve more complex problems, especially in solving systems of linear equations or working with computer graphics. Thus, mastering these operations is an important milestone in understanding linear algebra.

• Vector spaces and transformations give context to matrix multiplication.
• Applications range from robotic movements, 3D modeling, to quantum mechanics.

Step-by-Step Solution

Approaching matrix multiplication with a steady step-by-step method is key to ensuring accuracy and understanding. Let's recap and expand upon the provided solution for computing \(ABC\).

**Step 1: Calculate \(AB\)**
For the matrix \(AB\):

• The element in the first row, first column results from multiplying the first row of \(A\) by the first column of \(B\). Calculating: \((-1 \cdot 2) + (0 \cdot (-1)) = -2\)
• Proceed similarly for the remaining elements, carefully multiplying and summing appropriately.

This results in \( AB = \begin{bmatrix} -2 & -3 \ 0 & 5 \end{bmatrix} \).

**Step 2: Calculate \((AB)C\)**
Next, we multiply the resulting matrix \(AB\) by \(C\). Follow the same step-by-step multiplication method:

• The first element of \(ABC\) is \((-2 \cdot 1) + (-3 \cdot 0) = -2\).
• Continue until each element in \((AB)C\) is obtained.

The final result is \( ABC = \begin{bmatrix} -2 & -1 \ 0 & -5 \end{bmatrix} \).
This structured approach not only ensures the correct solution but builds a solid foundation for future, more complex problem solving.