Find the dimension of each matrix. Identify any square, column, or row matrices. [ − 4 8 2 3 ] \left[ \begin{matrix} -4 & 8 \\ 2 & 3 \end{matrix} \right]

Hello, everyone. We are asked to find the dimension of the matrix and categorize the matrix as a row column or square matrix. We have a matrix that has four elements two in one row two. In another, our answer choices are a, the dimension of the matrix is three by two B. The dimension of the matrix is two by three C. The dimension of the matrix is two by one and it's a column matrix D. The dimension of the matrix is two by two and it is a square matrix. First recall that matrices are labeled as the number of rows by the number of columns. Rows are horizontal columns are vertical. Here we have two horizontal rows and we also have two vertical columns. So the dimension of this matrix is a two by two. Is this a row matrix? No, as it has more than one row. Is it a column matrix? No, it has more than one column. Is it a square matrix? Yes. As both rows and columns equal each other. So this is answer choice D, the dimension of the matrix is two by two and it is a square matrix have a nice day.

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Problem 7

Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.
$[\begin{matrix} - 4 & 8 \\ 2 & 3 \end{matrix}]$

Verified step by step guidance

Step 1: Understand that the dimension of a matrix is given by the number of rows and columns it contains, usually written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

Step 2: Count the number of rows in the matrix. Each horizontal line of elements represents one row.

Step 3: Count the number of columns in the matrix. Each vertical line of elements represents one column.

Step 4: Write the dimension as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

Step 5: Identify the type of matrix based on its dimensions: if $m = n$, it is a square matrix; if $n = 1$, it is a column matrix; if $m = 1$, it is a row matrix.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Dimension

The dimension of a matrix is described by the number of its rows and columns, written as 'rows × columns'. For example, a matrix with 3 rows and 4 columns has a dimension of 3×4. Understanding dimensions is essential for identifying matrix types and performing operations.

Square Matrix

A square matrix has the same number of rows and columns (n×n). This property is important because square matrices have unique characteristics, such as the possibility of having a determinant and an inverse, which are not defined for non-square matrices.

Row and Column Matrices

A row matrix has only one row and multiple columns (1×n), while a column matrix has one column and multiple rows (m×1). Recognizing these helps in understanding matrix structure and is useful in vector representation and matrix multiplication.

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Textbook Question

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$