Question

Compute the following matrix products. a. \(\left[\begin{array}{rr}1 & 3 \\ 0 & -2\end{array}\right]\left[\begin{array}{rr}2 & -1 \\ 0 & 1\end{array}\right]\) b. \(\left[\begin{array}{rrr}1 & -1 & 2 \\ 2 & 0 & 4\end{array}\right]\left[\begin{array}{rrr}2 & 3 & 1 \\ 1 & 9 & 7 \\ -1 & 0 & 2\end{array}\right]\) c. \(\left[\begin{array}{rrr}5 & 0 & -7 \\ 1 & 5 & 9\end{array}\right]\left[\begin{array}{r}3 \\ 1 \\ -1\end{array}\right]\) d. \(\left[\begin{array}{lll}1 & 3 & -3\end{array}\right]\left[\begin{array}{rr}3 & 0 \\ -2 & 1 \\ 0 & 6\end{array}\right]\)e. \(\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{rr}3 & -2 \\ 5 & -7 \\ 9 & 7\end{array}\right]\) f. \(\left[\begin{array}{lll}1 & -1 & 3\end{array}\right]\left[\begin{array}{r}2 \\ 1 \\ -8\end{array}\right]\) g. \(\left[\begin{array}{r}2 \\ 1 \\\ -7\end{array}\right]\left[\begin{array}{lll}1 & -1 & 3\end{array}\right]\) h. \(\left[\begin{array}{ll}3 & 1 \\ 5 & 2\end{array}\right]\left[\begin{array}{rr}2 & -1 \\ -5 & 3\end{array}\right]\) i. \(\left[\begin{array}{lll}2 & 3 & 1 \\ 5 & 7 & 4\end{array}\right]\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]\) j. \(\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]\left[\begin{array}{ccc}a^{\prime} & 0 & 0 \\ 0 & b^{\prime} & 0 \\ 0 & 0 & c^{\prime}\end{array}\right]\)

Short answer

Solutions vary for each product, as shown in the detailed steps.

Step-by-step solution

Identify Matrices Multiplication Validity

We can multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix. Let's verify and find the dimensions after multiplication for each pair.

Multiply Matrices for Each Case

Apply matrix multiplication rules. For example, for two matrices \(A\) and \(B\), element \((i,j)\) in product \(C\) is \(C_{ij} = \sum_k A_{ik}B_{kj}\).

Key concepts

Matrix Dimensions

Matrix dimensions are the fundamental characteristics that define the shape of a matrix. They are written in terms of rows and columns. For instance, a matrix with 3 rows and 2 columns is described as a 3x2 matrix. When dealing with matrix multiplication, understanding dimensions is crucial.

Every matrix is not the same; their dimensions vary and can be described as:

• Row Matrix: A matrix with only one row, like (1xN).
• Column Matrix: A matrix with only one column, like (Nx1).
• Square Matrix: A matrix with the same number of rows and columns, like (NxN).
• Rectangular Matrix: A matrix where rows and columns differ, like (MxN).

Matrix multiplication is only possible when the inner dimensions match, which leads us to matrix multiplication rules.

Matrix Multiplication Rules

Matrix multiplication follows very specific rules. To multiply two matrices, say matrix A and matrix B, the number of columns in matrix A must be equal to the number of rows in matrix B.

Here is how the multiplication works:

• Identify the dimensions: If A is of dimensions (MxN) and B is (NxP), multiplication is possible, resulting in a matrix of dimensions (MxP).
• Compute each element: The element at the position (i,j) in the resulting matrix is computed as the sum of the products of corresponding elements from the row of the first matrix and the column of the second matrix.
• The simple formula used is: \(C_{ij} = \sum_{k} A_{ik} \cdot B_{kj}\)

These rules ensure that matrix multiplication is possible only under certain conditions, maintaining consistency and structure in mathematical operations. Ensuring the matrices conform to these rules allows the operations to yield meaningful results.

Matrix Operations

Matrix operations encompass more than just multiplication. They involve various other arithmetic operations, but multiplication is among the most complex due to its conditions and computations.

Let's touch upon other fundamental operations:

• Addition/Subtraction: Two matrices can be added or subtracted if, and only if, they have the same dimensions. Each corresponding element is added or subtracted.
• Scalar Multiplication: Every element of the matrix is multiplied by a scalar (a single number), irrespective of its dimensions.
• Transpose: Flipping a matrix over its diagonal to swap rows with columns. If a matrix is (MxN) then its transpose is (NxM).

Understanding each type of matrix operation broadens the scope of what can be achieved with matrices and develops a robust foundation for tackling more complex problems, such as solving systems of linear equations using matrices.