Introduction to matrices

The total elements are (3\times4=12). The number of elements is found by multiplying rows and columns.

In the notation \(a_{ij}\), the first index \(i\) denotes the row number (row index) and the second index \(j\) denotes the column number (column index). For example \(a_{21}\) is the element in row 2, column 1. Thus option B is correct. Option A is wrong because the column index is \(j\), option C is incorrect because the total number of elements is \(2\times3=6\) which is not what \(i\) denotes, and option D is wrong because the matrix name is \(A\). Exam tip: remember the convention a_{ij} → i = row, j = column (first index = row).

In (a_{23}), (2) is the row number and (3) is the column number. So it is the element in the (2)nd row and (3)rd column.

The governing definition of an identity matrix is a square matrix whose main-diagonal entries are 1 and whose off-diagonal entries are 0. In the given 2 × 2 matrix, both diagonal entries are 1, while the two entries away from the main diagonal are 0. Therefore it is the 2 × 2 identity matrix, usually written I₂, so option B is correct. It is not a zero matrix because its diagonal entries are not zero. It is not a row matrix because it has two rows, and although every square matrix is technically rectangular in a broad sense, “rectangular matrix” in this question is not the specific classification represented by the displayed form.

In a square matrix, the number of rows and columns is equal. In (4\times4), both numbers are (4).

The main diagonal has equal elements (5), and the remaining elements are (0). Such a diagonal matrix is called a scalar matrix.

By definition \(a_{ij}\) denotes the element in the i-th row and j-th column. So \(a_{12}\) is the element in the first row and second column. In the given matrix that entry is 2. Option A (1) is \(a_{11}\), while C (3) and D (4) are elements of the second row, so they are incorrect. Exam tip: read \(a_{ij}\) as “i-th row, j-th column” and locate row first, then column.

By notation \(a_{ij}\) denotes the element in the i-th row and j-th column. Thus \(a_{21}\) refers to the element in row 2, column 1. In the given matrix that entry is 9, so the correct answer is 9. For reference, 6 is \(a_{11}\) (row 1, column 1) and 7 is \(a_{12}\), so those options are incorrect. Exam tip: always read the subscript as (row, column) — i first, j second.

For two matrices to be equal, their order must be the same. Also, every corresponding element must be equal.

The rule given is \(a_{ij}=i+j\). For \(a_{22}\), the row index is \(i=2\) and the column index is \(j=2\), so \(a_{22}=2+2=4\). Option B (3) is incorrect — it would result from misreading indices as \(i=1,j=2\). Option D (5) is a simple arithmetic mistake. Exam tip: first identify the values of \(i\) and \(j\) clearly, then substitute into the formula.

Core concept: In \(a_{ij}\), \(i\) denotes the row and \(j\) the column. For \(a_{31}\) we have \(i=3\), \(j=1\). Substitute into the formula: \(a_{31}=2(3)-1=6-1=5\). Thus the correct value is 5. Closest distractor: option B (4) may arise if someone mistakenly used \(j=2\) or made an arithmetic slip. Exam tip: always check the order of indices (row, column) and substitute them into the formula before simplifying.

In (a_{ij}=i), all elements in a row equal the row number. The first row has (1) and the second row has (2).

By definition \(a_{ij}=j\) means each entry equals its column index. In a 2×2 matrix all entries in column 1 are 1 and all entries in column 2 are 2. Thus \(A=\begin{bmatrix}1&2\\1&2\end{bmatrix}\). Option B would correspond to \(a_{ij}=i\) (row index) and is therefore incorrect; options C and D do not follow the column-index rule either. Exam tip: write down the indices \(i,j\) for each position and substitute the rule to avoid sign/index errors.

In a diagonal matrix, all elements outside the main diagonal are (0). Main diagonal elements may be (0) or nonzero.

A transpose is formed by changing every row of a matrix into a column and every column into a row. The entries themselves do not change; only their positions change. This operation also changes the order of the matrix from rows by columns to columns by rows. It is written with the superscript symbol \(T\), so the required matrix is \(A^T\). This idea is useful whenever rows and columns must be compared or rearranged.

In the given matrix, the first row is \(1,2\), and the second row is \(3,4\). After transposing, the first column becomes \(1,3\), while the second column becomes \(2,4\). Therefore, \(A^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}\), which is option A. Option C is the original matrix, not its transpose.

On taking transpose, rows and columns interchange. Therefore the transpose of (2\times3) has order (3\times2).

Core idea: The main diagonal consists of entries where the row and column indices are equal (i.e. \(i=i\)), running from top-left to bottom-right. For \(A=\begin{bmatrix}1&0\\0&2\end{bmatrix}\), we have \(a_{11}=1\) and \(a_{22}=2\), so the diagonal elements are 1 and 2. The closest distractor D (0, 2) is wrong because its first entry 0 is \(a_{12}\), not a diagonal entry. Exam tip: Check positions with equal row and column indices (\(a_{11},a_{22},a_{33},\dots\)) to find the main diagonal quickly.

The main diagonal entries are those with equal row and column indices (a11, a22); here they are 0 and 0. Off-diagonal entries have unequal indices (a12 and a21). In this matrix a12 = 5 and a21 = 7, so the off-diagonal elements are 5 and 7. Closest distractors (options C or D) mix a diagonal zero with an off-diagonal value, which is why they are incorrect. Exam tip: locate positions with i=j first to find the main diagonal; all other positions (i≠j) are off-diagonal.

A matrix is a rectangular arrangement of numbers or expressions. It is written in rows and columns.

In (a_{13}), (1) tells the row and (3) tells the column. The index order is always row then column.