Introduction to matrices

The direct answer is B: the matrix has 12 elements. A matrix is a rectangular arrangement of numbers or other entries. If it has m rows and n columns, each row has n positions, and there are m rows, so the total number of entries is \(m\times n\). Here there are 3 rows and 4 columns, hence \(3\times4=12\). Option A, 7, comes from adding 3 and 4, but addition does not count all row-column positions. Option B, 12, is correct because every one of the 3 rows contains 4 elements. Option C, 1, would describe only a one-entry matrix, not a 3 by 4 matrix. Option D, 34, is not the product, sum, or valid count of the positions; the digits 3 and 4 do not combine by writing them side by side. You can also count row by row: first row has 4, second has 4, and third has 4, giving \(4+4+4=12\). Exam cue: for an \(m\times n\) matrix, multiply rows by columns, not add them.

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).

\(A\) has the form \(\begin{bmatrix}5&0\\0&5\end{bmatrix}=5\begin{bmatrix}1&0\\0&1\end{bmatrix}=5I\). When a diagonal matrix has the same non-zero value on all diagonal entries (so it equals a constant times the identity matrix), it is a scalar matrix. Hence \(A\) is 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 \(2\times 3\) matrix, \(i\) can be \(1\) (first row) or \(2\) (second row). Since \(a_{ij}=i\), every entry in the first row equals \(1\), and every entry in the second row equals \(2\). So \(A=\begin{bmatrix}1&1&1\\2&2&2\end{bmatrix}\).

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.

Transpose means rows become columns. So the entry at position \((1,2)\) becomes \((2,1)\), etc. For \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), we get \(A^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}\).

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.

Off-diagonal elements are the entries not lying on the main diagonal. In \(\begin{bmatrix}0&5\\7&0\end{bmatrix}\), the main diagonal entries are \(0\) (top-left) and \(0\) (bottom-right). Therefore, the off-diagonal elements are \(5\) and \(7\).

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.

Rows of a matrix are the horizontal lists of entries. For this 2×3 matrix the first row is (1,2,3) and the second (bottom) row is (4,5,6). Option A is the first row; options C and D are columns (first and third column respectively). Exam tip: use the row index (the first index) to identify which horizontal list is the required row.

A column of a matrix is the vertical list of entries. In the given matrix the third column takes the third entry from the first row (3) and the third entry from the second row (6), so it is \(\begin{pmatrix}3\\6\end{pmatrix}\). The closest distractor \(\begin{pmatrix}2\\5\end{pmatrix}\) is the second column, and \(\begin{pmatrix}4\\5\\6\end{pmatrix}\) has three entries and does not match the 2×3 matrix. Exam tip: read column entries top to bottom and use the column index to pick the correct vertical list.

When all elements are (0), the matrix is a zero matrix. This is a direct definition-based question.

(i=j) represents the main diagonal, where the elements are (1). For (i\ne j), the other elements are (0), so it is an identity matrix.

\(a_{ij}\) denotes the element in the i-th row and j-th column. Thus \(a_{22}\) is the element in the second row and second column. In matrix \(A\) that entry is \(6\). The other choices correspond to \(9=a_{11},\ 8=a_{12},\ 7=a_{21}\). Exam tip: Always read \(a_{ij}\) as row i, column j to locate the element quickly.