Q.A square matrix where every element is unity is called an identity matrix.

The statement is false. Here’s why:

1. A square matrix where every element is unity (all entries equal to 1) is called a unit matrix or a matrix of ones, often denoted by $J$.
2. The identity matrix, denoted $I$, has 1s only on the main diagonal and 0s elsewhere: $I_{ij} = 1$ if $i = j$, and $0$ otherwise. …

The Concept: What "Identity" Actually Means

The word "identity" in mathematics refers to something that leaves other things unchanged when combined with them. For matrices, the identity matrix $I_n$ (of size $n \times n$) is the multiplicative identity: for any matrix $A$ of compatible size, $A I = A$ and $I A = A$. This property is what makes it special.

A matrix where every element is 1 — let's call it $J_n$ — does not have this property. Multiply $J$ by another matrix, and you generally get something completely different, not the original matrix back.

Step-by-Step Reasoning

1. Define the two matrices clearly. The $n \times n$ identity matrix $I_n$ has entries:

The $n \times n$ matrix of all ones $J_n$ has entries:

2. Test the defining property of an identity matrix.

   Take any $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

   • Multiply by $I_2$: $A I_2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = A$. Works.
   • Multiply by $J_2$: $A J_2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} a+b & a+b \\ c+d & c+d \end{pmatrix}$, which is not equal to $A$ unless $b=0$ and $a=c+d$, etc. So $J$ fails the identity test.

3. Check the name. …

Method: Verifying a Named-Matrix Definition Against Its Defining Property

Use this for any statement of the form "a matrix that looks like ___ is called ___." Names of special matrices (identity, zero, diagonal, scalar) are tied to a defining property, not merely to how the entries look.

Steps

Step 1: Write down the exact definition the name carries.

The identity matrix $I_n$ is defined by its action: $AI=IA=A$ for every conformable $A$. Its entry pattern is $1$ on the main diagonal, $0$ elsewhere:

Step 2: Write down what the described matrix actually is. …

Common Mistakes

Mistake 1: Confusing "all entries are 1" with "diagonal entries are 1."

Why it's wrong: the identity matrix has $1$s only on the main diagonal and $0$s elsewhere; a matrix of all ones is completely filled. Correct approach: recall $I$ is mostly zeros — sparse, not dense.

Mistake 2: Naming a matrix from its appearance instead of its property.

Why it's wrong: "identity" refers to the property $AI=IA=A$, not to a specific-looking grid. Correct approach: test the action — an all-ones matrix $J$ gives $AJ\neq A$ in general, so it fails the identity test. …