Find the dimension of each matrix. Identify any square, column, or row matrices. [−962418]\left[ \begin{matrix} -9 & 6 & 2 \\ 4 & 1 & 8 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 8

Chapter 6, Problem 8

Find the dimension of each matrix. Identify any square, column, or row matrices.
$[\begin{matrix} - 9 & 6 & 2 \\ 4 & 1 & 8 \end{matrix}]$

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Recall that the dimension of a matrix is given by the number of rows and columns it has, usually written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

To find the dimension of each matrix, count the number of horizontal entries (rows) and the number of vertical entries (columns) in the matrix.

Identify if the matrix is a square matrix by checking if the number of rows equals the number of columns, i.e., if $m = n$.

Check if the matrix is a column matrix by verifying if it has exactly one column, i.e., if $n = 1$ regardless of the number of rows.

Check if the matrix is a row matrix by verifying if it has exactly one row, i.e., if $m = 1$ regardless of the number of columns.

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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 the dimension 3×4. Knowing the dimension helps in understanding the structure and possible operations on the matrix.

Square Matrix

A square matrix has the same number of rows and columns, such as 2×2 or 5×5. This type of matrix is important because it allows for special operations like finding determinants and inverses, 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). These matrices are useful in representing vectors and have unique properties in matrix multiplication and transformations.

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Textbook Question

Solve each system by substitution.

4x + 3y = -13

-x + y = 5

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Use the given row transformation to change each matrix as indicated.

$[\begin{matrix} 1 & 4 \\ 4 & 7 \end{matrix}],$

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Textbook Question

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.

6x + 10y = -11

9x + 6y = -3

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Textbook Question

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.

2x² = 3y + 23

y = 2x - 5

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 5/(3x(2x + 1))

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Textbook Question

Use the given row transformation to change each matrix as indicated.

$[\begin{matrix} 1 & - 4 \\ 7 & 0 \end{matrix}],$