Problem 39 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{rr}{-10} & {20} \\ {5} & {25}\end{array}\right], B=\left[\begin{array}{rr}{40} & {10} \\ {-20} & {30}\end{array}\right], C=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1} \\ {1} & {0}\end{array}\right] $$ $$ (A B)^{2} $$

Step-by-Step Solution

Answer

The result of \((AB)^2\) is \(\left[\begin{array}{rr}{490000} & {0} \\ {0} & {490000}\end{array}\right]\)."

Step 1: Check Matrix Multiplication Compatibility

First, check if matrices \(A\) and \(B\) can be multiplied. Matrix \(A\) is a \(2 \times 2\) matrix and so is \(B\). Because the number of columns in \(A\) (2 columns) matches the number of rows in \(B\) (2 rows), the multiplication is possible.

Step 2: Perform the Multiplication A B

Calculate the product of matrices \(A\) and \(B\):\[ A \cdot B = \left[\begin{array}{rr}{-10} & {20} \ {5} & {25}\end{array}\right] \cdot \left[\begin{array}{rr}{40} & {10} \ {-20} & {30}\end{array}\right] \]The resulting matrix will be of size \(2 \times 2\). Calculate each element by performing the dot product of rows of \(A\) with columns of \(B\).

Step 3: Calculate Elements of A B

\[(1,1) = (-10)*40 + 20*(-20) = -400 - 400 = -800\]\[(1,2) = (-10)*10 + 20*30 = -100 + 600 = 500\]\[(2,1) = 5*40 + 25*(-20) = 200 - 500 = -300\]\[(2,2) = 5*10 + 25*30 = 50 + 750 = 800\]Thus, \(A \cdot B = \left[\begin{array}{rr}{-800} & {500} \ {-300} & {800}\end{array}\right]\).

Step 4: Calculate (AB)^2

We have \(AB\), now calculate \[(AB)^2 = (AB) \cdot (AB)\]\[ AB = \left[\begin{array}{rr}{-800} & {500} \ {-300} & {800}\end{array}\right]\]The calculation is similar to the previous step: compute the product by considering the dot product of rows and columns.

Step 5: Calculate Elements of (AB)^2

Calculate each element of \((AB)\cdot(AB)\): \[(1,1) = (-800)*(-800) + 500*(-300) = 640000 - 150000 = 490000\]\[(1,2) = (-800)*500 + 500*800 = -400000 + 400000 = 0\]\[(2,1) = (-300)*(-800) + 800*(-300) = 240000 - 240000 = 0\]\[(2,2) = (-300)*500 + 800*800 = -150000 + 640000 = 490000\]Thus, \((AB)^2 = \left[\begin{array}{rr}{490000} & {0} \ {0} & {490000}\end{array}\right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra. It allows you to combine two matrices, producing another matrix as the result. To perform matrix multiplication, the number of columns in the first matrix must match the number of rows in the second. This ensures the matrices are compatible for multiplication.

• For example, if matrix \(A\) is a \(2 \times 2\) matrix and matrix \(B\) is also a \(2 \times 2\) matrix, you can multiply them.
• The resulting matrix from multiplying \(A\) and \(B\) will also be \(2 \times 2\).

To calculate each entry in the resulting matrix, take the dot product of the corresponding row of the first matrix with the column of the second. This means multiplying matching elements and summing the products. Matrix multiplication is not commutative, meaning \(A \cdot B\) does not necessarily equal \(B \cdot A\).
To master matrix multiplication, practice with different sizes and ensure the matrices' dimensions align correctly.

Matrix Squaring

Matrix squaring is a specific type of matrix multiplication where a matrix is multiplied by itself. If you have a square matrix, say \(A\), you can square it by computing \(A \cdot A\). When dealing with squaring, it’s crucial the matrix is square, meaning it has the same number of rows and columns.

• For instance, for a \(2 \times 2\) matrix \(AB\), you can find \((AB)^2\) by multiplying \(AB\) by itself.
• This results in a matrix of the same dimension: \(2 \times 2\).

Matrix squaring retains the properties of matrix multiplication, so each element in the resulting matrix requires calculating the dot product of rows and columns of the matrix. This powerful operation is used in various mathematical fields, including systems of equations and transformations.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. Matrices are essential elements in linear algebra, used to represent and solve linear equations and transformations. A fundamental part of linear algebra is understanding how matrix operations like multiplication impact vectors within a vector space.

• Matrices can represent systems of linear equations, where each row might correspond to an equation.
• Operations such as multiplication and squaring play a significant role in changing perspectives and transforming these vectors.

Linear algebra is crucial for various applications in science and engineering, where it facilitates the modeling and solving of real-world problems. It helps understand spaces in multiple dimensions and transform data efficiently.

Matrix Compatibility Check

Performing a matrix compatibility check is a vital step before any matrix operation. It involves ensuring that matrices involved are of suitable sizes for operations like multiplication or addition. This step guarantees accurate and feasible results.

• To multiply two matrices, ensure the number of columns in the first matrix matches the number of rows in the second.
• Checking compatibility prevents errors in calculations and operations.

This check involves examining the dimensions of the matrices and is crucial for determining whether operations like multiplication are possible. For instance, if matrix \(A\) is \(2 \times 3\), it can only be multiplied by a matrix that is \(3 \times n\), where \(n\) is any positive integer. Skipping compatibility checks can lead to undefined mathematical operations, which is why it's a foundational step in working with matrices.