Class 12 • mathematics • Chapter 3
Matrices
True & False Quiz
Rows. Columns. Order.
✓True
✗False
25
Questions
|
Ch.3
Chapter
|
XII
Class
Why True & False for Matrices?
How this format sharpens your conceptual clarity
🔵 Matrices give a compact way to represent and solve linear systems, transformations, and networks.
✅ T/F targets matrix multiplication order, symmetric/skew-symmetric decomposition, and when operations are even defined.
🎯 Trap: matrix multiplication is generally NOT commutative (AB ≠ BA), even when both products are defined.
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
For a matrix \(A=[a_{ij}]_{m\times n}\), the element \(a_{ij}\) lies in the \(i\)-th row and \(j\)-th column.
Q 2
A matrix having exactly one row is called a row matrix.
Q 3
A square matrix must have the same number of rows and columns.
Q 4
Two matrices of different orders can be equal if all their corresponding entries are equal.
Q 5
The zero matrix of order \(3\times2\) has six zero entries.
Q 6
If \(A\) is a \(2\times3\) matrix and \(B\) is a \(3\times4\) matrix, then \(AB\) is defined and has order \(2\times4\).
Q 7
Matrix multiplication is commutative for all matrices for which both \(AB\) and \(BA\) are defined.
Q 8
If \(A\) and \(B\) are matrices of the same order, then \(A+B\) is defined.
Q 9
For any matrices \(A,B,C\) for which the operations are defined, \(A(B+C)=AB+AC\).
Q 10
For every square matrix \(A\), \(AI=IA=A\), where \(I\) is the identity matrix of the same order as \(A\).
Q 11
If \(AB=O\), then necessarily \(A=O\) or \(B=O\).
Q 12
If \(A\) is a square matrix and \(A^2=A\), then \(A\) is called an idempotent matrix.
Q 13
If \(A\) is a square matrix satisfying \(A^2=I\), then \(A^{-1}=A\).
Q 14
If \(A\) is a \(2\times3\) matrix, then \(A^T\) is a \(3\times2\) matrix.
Q 15
For any matrices \(A\) and \(B\) of compatible orders, \((A+B)^T=A^T+B^T\).
Q 16
For compatible matrices \(A\) and \(B\), \((AB)^T=A^TB^T\).
Q 17
A square matrix \(A\) is symmetric if \(A^T=A\).
Q 18
A square matrix \(A\) is skew-symmetric if \(A^T=-A\).
Q 19
Every diagonal matrix is symmetric.
Q 20
Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
Q 21
In a skew-symmetric matrix, every diagonal element must be zero.
Q 22
If \(A\) is invertible, then its inverse \(A^{-1}\) is unique.
Q 23
If \(A\) and \(B\) are invertible square matrices of the same order, then \(AB\) is invertible and \((AB)^{-1}=A^{-1}B^{-1}\).
Q 24
If \(A\) is invertible and \(AB=AC\), then \(B=C\).
Q 25
If \(A\) and \(B\) are square matrices of the same order and \(AB=BA\), then \((A+B)^2=A^2+2AB+B^2\).
Key Takeaways — Matrices
Core facts for CBSE Boards & JEE
1
Matrix addition requires identical order (rows × columns); multiplication AB needs columns of A = rows of B.
2
In general AB ≠ BA, even when both products exist and are of the same order.
3
Every square matrix A can be written as the sum of a symmetric and a skew-symmetric matrix: A = ½(A+Aᵀ) + ½(A−Aᵀ).
4
For a symmetric matrix, Aᵀ = A; for a skew-symmetric matrix, Aᵀ = −A, and all diagonal elements are 0.
5
(AB)ᵀ = BᵀAᵀ — the transpose of a product reverses the order.
6
A scalar matrix is a diagonal matrix with all diagonal entries equal; the identity matrix I is a special case with entries 1.