Problem 93 - Algebra and Trigonometry

Question

Where possible, find each product. a. \(\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\) b. \(\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rrr}{-1} & {0} & {1} \\ {0} & {-1} & {1}\end{array}\right]\)

Step-by-Step Solution

Answer

The results are: \n a. \[\left[\begin{array}{rr}{-1} & {0} \ {0} & {1} \end{array}\right]\]\n b. \[\left[\begin{array}{rrr}{1} & {0} & {-1} \ {0} & {1} & {-1} \end{array}\right]\]

Step 1: Matrix multiplication rules

In general, to multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Then, to find element \(C_{ij}\) of the result matrix \(C\), sum the products of the elements of the \(i\)th row of the first matrix and the \(j\)th column of the second matrix.

Step 2: Multiply matrices in part a

Here, both matrices are 2x2 and so the multiplication is valid. We obtain the result matrix by following the rule described above: \[ \left[\begin{array}{rr}{1*-1 + 0*0} & {1*0 + 0*-1} \ {0*-1 + -1*0} & {0*0 + -1*-1} \end{array}\right] = \left[\begin{array}{rr}{-1} & {0} \ {0} & {1} \end{array}\right] \]

Step 3: Multiply matrices in part b

The first matrix has dimensions 2x2 and the second matrix has dimensions 2x3, so their multiplication is valid and we will obtain a 2x3 matrix as a result. By applying the rule as in part a, we compute: \[ \left[\begin{array}{rrr}{-1*-1 + 0*0} & {-1*0 + 0*-1} & {-1*1 + 0*1} \ {0*-1 + -1*0} & {0*0 + -1*-1} & {0*1 + -1*1} \end{array}\right] = \left[\begin{array}{rrr}{1} & {0} & {-1} \ {0} & {1} & {-1} \end{array}\right] \]

Key Concepts

Product of Matrices

Matrix multiplication is a foundational concept in linear algebra. But not just any two matrices can be multiplied; there are certain conditions that must be fulfilled. The "product of matrices" refers to the outcome obtained when two matrices are multiplied together following specific rules. The central requirement for multiplying two matrices is that the number of columns in the first matrix must match the number of rows in the second matrix. This ensures that each element from a row in the first matrix has a corresponding element to pair with in the second matrix's column.
When these conditions are met, to calculate each element in the resulting matrix, we take the row elements from the first matrix and column elements from the second, multiply them together individually, and then sum those products. Each unique pairing of rows and columns gives us a single element in the new matrix.
Understanding this process is crucial as it forms the basis for more advanced matrix operations and helps in solving systems of linear equations.

2x2 Matrix

A 2x2 matrix is one of the simplest kinds of matrices, having two rows and two columns. This format is often used in basic exercises because it is straightforward while still allowing for the exploration of fundamental concepts like determinant and inverse.
A generic 2x2 matrix looks like this:

• First row: two numbers
• Second row: two numbers

When multiplying two 2x2 matrices, the resulting matrix will also be 2x2 if both matrices are compatible for multiplication. For example, consider matrices a = \[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\] and b = \[\begin{bmatrix}-1 & 0 \0 & -1\end{bmatrix}\]. Following the matrix multiplication rule results in a matrix \[\begin{bmatrix}-1 & 0 \0 & 1\end{bmatrix}\].
This example illustrates that even though we start with simple structures, applying matrix operations can lead us to insightful results.

2x3 Matrix

A 2x3 matrix is a bit more complex, consisting of two rows and three columns. This makes it suitable for slightly more intricate problems where the goal is usually to get a result that isn't square, such as in the case of a product that results in a non-square matrix due to differing dimensions.
Specifically, the format looks like this:

• First row: three numbers
• Second row: three numbers

When a 2x2 matrix is multiplied by a 2x3 matrix, the resulting matrix will have dimensions 2x3. This stems from the core principle that the outer dimensions of the matrices involved in multiplication determine the dimensions of the result.
Consider the equation where matrix b = \[\begin{bmatrix}-1 & 0 \0 & -1\end{bmatrix}\] is multiplied by a = \[\begin{bmatrix}-1 & 0 & 1 \0 & -1 & 1\end{bmatrix}\]. The result is a 2x3 matrix \[\begin{bmatrix}1 & 0 & -1 \0 & 1 & -1\end{bmatrix}\]. This showcases how different matrix dimensions interact during multiplication, illustrating the diversity of possible outcomes.

Matrix Dimensions

Matrix dimensions denote the number of rows and columns in a matrix, expressed as "rows by columns". Understanding dimensions is crucial because it dictates the compatibility of matrices for various operations, especially multiplication.
A matrix with dimensions "m x n" has "m" rows and "n" columns. This notation provides a quick way to determine whether matrix multiplication is feasible. If you have two matrices - the first being "a x b" and the second "b x c", multiplication can proceed, resulting in a new matrix with dimensions "a x c".

• The first matrix's columns must match the second matrix's rows for compatibility.
• The resulting matrix's dimensions are taken from the outer values of the original matrices, offering a straightforward way to predict the output's size.

For instance, if a 2x2 matrix is multiplied by a 2x3 matrix, the resulting dimensions will be 2x3. Recognizing the importance of dimensions not only enhances your problem-solving skills but also supports predictive analysis in mathematical problems. This understanding builds the foundation for exploring larger, more complex matrices in advanced studies.