Problem 44 - Algebra 2

Question

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ (D D) E $$

Step-by-Step Solution

Answer

The product (DD)E is undefined because after calculating DD, which is a 3x3 matrix, it cannot be multiplied by E, which is also a 3x3 matrix. The inner dimensions do not match.

Step 1: Calculate DxD (Matrix Multiplication)

To find the square of matrix D (denoted as DxD), multiply matrix D by itself. Use matrix multiplication rules by taking the row of the first matrix and multiplying it by the column of the second matrix, summing the products to produce the new elements.

Step 2: Multiply the Result by E (Matrix Multiplication)

After finding DxD, multiply the result by matrix E. Use the same matrix multiplication method as in Step 1, taking the rows from the DxD matrix and multiplying them by the columns of matrix E, summing the products to find the elements of the final matrix.

Step 3: Verify the Dimensions

Before performing matrix multiplication, verify the dimensions of the matrices to ensure that they can be multiplied. For matrices A (m x n) and B (p x q), the number of columns in A (n) must be equal to the number of rows in B (p) in order to define the multiplication. The resulting matrix will have dimensions m x q.

Key Concepts

Matrices Operations

Matrices are fundamental to a variety of calculations in mathematics and physics. Operations on matrices are defined by a set of rules that dictate how we can manipulate these structured arrays of numbers. The most common operations include matrix addition, subtraction, and multiplication. When adding or subtracting matrices, it’s crucial to match elements from each matrix that have the same position. This means that we can only add or subtract matrices if they have the same dimensions.
For multiplication, however, the rules are different and a bit more complex. It involves calculating the dot product between each row of the first matrix and each column of the second matrix. Due to this, not all matrices can be multiplied together, as their dimensions must align according to specific rules.

Dimension of Matrices

When we refer to the 'dimension' of a matrix, we're talking about the number of rows and columns that it contains — typically noted as 'm x n', where 'm' is the number of rows, and 'n' is the number of columns. This concept is essential because the possibility of performing certain matrix operations depends on the dimensions of the matrices involved.

• A matrix with dimensions 3 x 4 has 3 rows and 4 columns.
• A matrix dimension also indicates the size of the matrix.
• Only matrices with the same dimensions can be added or subtracted.

Having a clear understanding of matrix dimensions is necessary before engaging in matrix operations, as it determines whether these operations can be executed.

Matrix Multiplication Rules

Matrix multiplication is not as straightforward as addition or subtraction. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The resultant matrix has the dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. Here’s how it works in steps:

• Select a row from the first matrix.
• Select a column from the second matrix.
• Multiply the corresponding entries from the row and column.
• Sum these products to yield a single number, which is an element of the product matrix.

This process repeats for each row of the first matrix and each column of the second matrix until all positions in the result matrix are filled. Importantly, this operation is not commutative, meaning that the order of multiplication matters.

Undefined Matrix Operations

Some matrix operations might be 'undefined' because they don't meet the necessary conditions to be performed. In terms of matrix multiplication, this happens when the number of columns in the first matrix does not equal the number of rows in the second matrix. Here's what you need to remember:

• If you try to multiply a 2x3 matrix with a 4x2 matrix, the operation is undefined because the number of columns in the first matrix (3) doesn't match the number of rows in the second (4).
• Such an attempt will lead to a conceptual mismatch since there's no way to pair up elements of the respective row and column for multiplication.

It's critical to verify the dimensions before attempting matrix operations to avoid running into undefined scenarios. Doing so ensures that time is not wasted on impossible calculations and maintains the integrity of your mathematical work.