Q.Compute the following:

Concept: Matrix Addition — add corresponding entries.

(i)

Add entrywise:

Top-left: $a + a = 2a$

Top-right: $b + b = 2b$

Bottom-left: $-b + b = 0$

Bottom-right: $a + a = 2a$

(ii)

Add entrywise:

Top-left: $(a^2+b^2) + 2ab = (a+b)^2$

Top-right: $(b^2+c^2) + 2bc = (b+c)^2$

Bottom-left: $(a^2+c^2) + (-2ac) = (a-c)^2$

Bottom-right: $(a^2+b^2) + (-2ab) = (a-b)^2$

(iii)

Add entrywise:

Row 1: $[-1+12,\; 4+7,\; -6+6] = [11,\; 11,\; 0]$

Row 2: $[8+8,\; 5+0,\; 16+5] = [16,\; 5,\; 21]$

Row 3: $[2+3,\; 8+2,\; 5+4] = [5,\; 10,\; 9]$

(iv)

Add entrywise:

Top-left: $\cos^2 x + \sin^2 x = 1$

Top-right: $\sin^2 x + \cos^2 x = 1$

Bottom-left: $\sin^2 x + \cos^2 x = 1$

Bottom-right: $\cos^2 x + \sin^2 x = 1$

The Core Idea

Matrix addition is the simplest operation in linear algebra — you just add corresponding entries. If two matrices have the same number of rows and columns, their sum is a new matrix where each element is the sum of the elements in the same position. There is no cross-talk between positions, no multiplication, no special rules. The only requirement is that the matrices be of the same order.

Let’s apply this cleanly to each part.

(i) $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} + \begin{bmatrix} a & b \\ b & a \end{bmatrix}$

1. Both matrices are $2 \times 2$, so addition is defined.
2. Add the top-left entries: $a + a = 2a$.
3. Add the top-right entries: $b + b = 2b$.
4. Add the bottom-left entries: $-b + b = 0$.
5. Add the bottom-right entries: $a + a = 2a$.

The result is $\begin{bmatrix} 2a & 2b \\ 0 & 2a \end{bmatrix}$.

(ii) $\begin{bmatrix} a^2+b^2 & b^2+c^2 \\ a^2+c^2 & a^2+b^2 \end{bmatrix} + \begin{bmatrix} 2ab & 2bc \\ -2ac & -2ab \end{bmatrix}$

1. Both are $2 \times 2$, so proceed element-wise.
2. Top-left: $(a^2+b^2) + 2ab = a^2 + 2ab + b^2 = (a+b)^2$.
3. Top-right: $(b^2+c^2) + 2bc = b^2 + 2bc + c^2 = (b+c)^2$.
4. Bottom-left: $(a^2+c^2) + (-2ac) = a^2 - 2ac + c^2 = (a-c)^2$.
5. Bottom-right: $(a^2+b^2) + (-2ab) = a^2 - 2ab + b^2 = (a-b)^2$.

The result is $\begin{bmatrix} (a+b)^2 & (b+c)^2 \\ (a-c)^2 & (a-b)^2 \end{bmatrix}$.

(iii) $\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}$

1. Both are $3 \times 3$, so add each corresponding entry. It helps to work row by row.

First row:

• Column 1: $-1 + 12 = 11$
• Column 2: $4 + 7 = 11$
• Column 3: $-6 + 6 = 0$

Second row:

• Column 1: $8 + 8 = 16$
• Column 2: $5 + 0 = 5$
• Column 3: $16 + 5 = 21$

Third row:

• Column 1: $2 + 3 = 5$
• Column 2: $8 + 2 = 10$
• Column 3: $5 + 4 = 9$

The result is $\begin{bmatrix} 11 & 11 & 0 \\ 16 & 5 & 21 \\ 5 & 10 & 9 \end{bmatrix}$.

(iv) $\begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix} + \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}$

1. Both are $2 \times 2$, so add element-wise.
2. Top-left: $\cos^2 x + \sin^2 x = 1$ (using the Pythagorean identity).
3. Top-right: $\sin^2 x + \cos^2 x = 1$.
4. Bottom-left: $\sin^2 x + \cos^2 x = 1$.
5. Bottom-right: $\cos^2 x + \sin^2 x = 1$.

Every entry becomes $1$, so the result is $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$.

Method: Entrywise Addition, Including Symbolic and Trigonometric Entries

Use this for sums of matrices whose entries are algebraic expressions or trig functions. The rule is still entrywise, but the arithmetic per entry may simplify using an identity.

Steps

Step 1: Confirm equal orders and add corresponding entries.

$(A+B)_{ij}=a_{ij}+b_{ij}$, done position by position for matrices of the same order.

Step 2: Simplify each combined entry algebraically.

Look for perfect-square patterns, e.g. $a^2+b^2+2ab=(a+b)^2$ and $a^2+c^2-2ac=(a-c)^2$; respect the sign attached to each cross term.

Step 3: Apply standard identities where entries are trigonometric.

Use identities such as $\sin^2\theta+\cos^2\theta=1$ to collapse entries. Present each simplified entry in its position to get the final matrix.

Common Mistakes

Mistake 1: Mishandling the sign of a cross term when factoring.

Why it's wrong: $a^2+c^2+(-2ac)=(a-c)^2$, not $(a+c)^2$; the negative sign flips the binomial. Correct approach: keep the given sign of the $2ac$-type term before factoring.

Mistake 2: Forgetting to use the Pythagorean identity to simplify.

Why it's wrong: leaving $\sin^2\theta+\cos^2\theta$ unsimplified misses that each such entry is just $1$. Correct approach: apply $\sin^2\theta+\cos^2\theta=1$ to reduce entries.

Mistake 3: Adding matrices of different orders.

Why it's wrong: addition is undefined unless the orders match. Correct approach: verify both matrices share the same order before adding.