Problem 45 - Algebra 2

Question

What are the dimensions of the matrix that results from the multiplication shown? \(\left[\begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {f} \\ {g} & {h} & {i} \\ {j} & {k} & {l}\end{array}\right] \cdot\left[\begin{array}{l}{7} \\\ {4} \\ {6}\end{array}\right]\) A. \(1 \times 4\) B. \(3 \times 3\) C. \(4 \times 1\) D. \(4 \times 3\)

Step-by-Step Solution

Answer

C. The dimension is \(4 \times 1\).

Step 1: Identify the Dimensions of Matrices

The first matrix has 4 rows and 3 columns, making it a \(4 \times 3\) matrix. The second matrix has 3 rows and 1 column, making it a \(3 \times 1\) matrix.

Step 2: Check Multiplicability Condition

For matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Here, both are 3, so multiplication is possible.

Step 3: Determine the Dimensions of the Product Matrix

The resulting matrix after multiplication will have dimensions equal to the number of rows from the first matrix and the number of columns from the second matrix. Therefore, the resulting dimension is \(4 \times 1\).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is key to performing matrix multiplication. Each matrix is characterized by its dimensions, which are defined by the number of rows and columns it contains. For example, a matrix with 4 rows and 3 columns is called a "4 by 3" matrix, often noted as \(4 \times 3\). The dimensions of a matrix not only describe its size and shape but also play a critical role in determining whether two matrices can be multiplied together.When working with matrices, it's essential to correctly identify their dimensions, as these will dictate the steps and feasibility of further matrix operations, such as calculating the matrix product.

Matrix Product

The matrix product, or matrix multiplication, involves two matrices where elements from the rows of the first matrix are combined with elements from the columns of the second matrix. Matrix multiplication is not as straightforward as simple element-wise multiplication. To perform matrix multiplication, one must ensure that the number of columns in the first matrix matches the number of rows in the second matrix.Example: If we have a \(4 \times 3\) matrix multiplied by a \(3 \times 1\) matrix, we confirm that multiplication is possible because the number of columns in the first matrix (3) is equal to the number of rows in the second matrix (3).The resulting matrix will have dimensions defined by the number of rows from the first matrix and the number of columns from the second matrix. In this instance, the matrix product will be a \(4 \times 1\) matrix.

Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors, vector spaces, linear mappings, and systems of linear equations. Matrices are a fundamental concept in linear algebra, acting as tools to represent and solve various mathematical problems. Matrix multiplication is one of the core operations within linear algebra, allowing transformations that are difficult to visualize without algebraic representation. For example, matrices can perform rotations, scaling, and translations of geometric figures, simplifying complex calculations. Understanding matrices and their operations provides foundational skills used in numerous applications, from solving linear equations to modeling networks and even in advanced fields like quantum mechanics.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, scalar multiplication, and finding determinants and inverses. Each operation follows specific rules that leverage the arrangement and dimensions of matrices. Key Operations:

• Addition/Subtraction: Only matrices of the same dimensions can be added or subtracted.
• Scalar Multiplication: Each entry of the matrix is multiplied by a constant scalar value.
• Multiplication: The focus here, requires the number of columns in the first matrix to match the number of rows in the second matrix.
• Determinant: Applicable to square matrices, offering insight into properties like invertibility.
• Inverse: A matrix inverse exists only for square matrices and acts much like division.

Understanding these operations enhances the ability to solve systems of equations and perform transformations, crucial in disciplines such as computer graphics, engineering, and data science.