a. Give the order of each matrix.


b. If $A=\left[a_{ij}\right]$, identify $a_{32}$ and $a_{23}$, or explain why identification is not possible.
$\begin{bmatrix}4 & -7 & 5 \\-6 & 8 & -1\end{bmatrix}$

Verified step by step guidance

1. Step 1: Identify the order of the matrix by counting the number of rows and columns. The matrix has 2 rows and 3 columns, so its order is \$2 \times 3$.
2. Step 2: Understand the notation \(a_{ij}\), where \(i\) represents the row number and \(j\) represents the column number of the element in the matrix.
3. Step 3: To find \(a_{32}\), look for the element in the 3rd row and 2nd column. Since the matrix has only 2 rows, \(a_{32}\) does not exist.
4. Step 4: To find \(a_{23}\), look for the element in the 2nd row and 3rd column. This element is \(-1\).
5. Step 5: Summarize the findings: The order of the matrix is \$2 \times 3\(, \)a_{32}\( is not defined because the matrix has only 2 rows, and \)a_{23} = -1$.

Key Concepts

Matrix Order

The order of a matrix is defined by the number of its rows and columns, expressed as 'rows × columns'. For example, a matrix with 2 rows and 3 columns has an order of 2 × 3. Knowing the order helps in identifying the size and structure of the matrix.

Matrix Elements and Notation

Each element in a matrix is denoted by a_ij, where 'i' represents the row number and 'j' the column number. For instance, a_32 refers to the element in the 3rd row and 2nd column. This notation helps in precisely identifying and working with specific entries in the matrix.

Index Validity in Matrices

When identifying elements like a_32 or a_23, it is essential to verify if the specified row and column indices exist within the matrix's order. If the matrix has fewer rows or columns than the indices suggest, the element cannot be identified. This ensures accurate referencing within the matrix.