Matrix Magic - Fun with Matrices!

Problem 1: Matrix Addition

Given matrices:

A = ⎡1 9⎤
⎢3 4⎥
⎣8 -3⎦

B = ⎡5 7⎤
⎢3 3⎥
⎣1 0⎦

Verify that:

(i) A + B = B + A

(ii) A + (-A) = (-A) + A = O (zero matrix)

Part (i): A + B = B + A

Let's calculate A + B:

A + B = ⎡1+5 9+7⎤
⎢3+3 4+3⎥
⎣8+1 -3+0⎦

= ⎡6 16⎤
⎢6 7⎥
⎣9 -3⎦

Now calculate B + A:

B + A = ⎡5+1 7+9⎤
⎢3+3 3+4⎥
⎣1+8 0-3⎦

= ⎡6 16⎤
⎢6 7⎥
⎣9 -3⎦

We can see that A + B = B + A! ✔️

Part (ii): A + (-A) = (-A) + A = O

First, find -A:

-A = ⎡-1 -9⎤
⎢-3 -4⎥
⎣-8 3⎦

Now calculate A + (-A):

A + (-A) = ⎡1-1 9-9⎤
⎢3-3 4-4⎥
⎣8-8 -3+3⎦

= ⎡0 0⎤
⎢0 0⎥
⎣0 0⎦

Similarly, (-A) + A gives the same zero matrix.

Verified! A + (-A) = (-A) + A = O ✔️

Problem 2: Matrix Addition with Three Matrices

Given matrices:

A = ⎡4 3 1⎤
⎢2 3 -8⎥
⎣1 0 -4⎦

B = ⎡2 3 4⎤
⎢1 9 2⎥
⎣-7 1 -1⎦

C = ⎡8 3 4⎤
⎢1 -2 3⎥
⎣2 4 -1⎦

Verify that: A + (B + C) = (A + B) + C

Step 1: Calculate B + C

B + C = ⎡2+8 3+3 4+4⎤
⎢1+1 9-2 2+3⎥
⎣-7+2 1+4 -1-1⎦

= ⎡10 6 8⎤
⎢2 7 5⎥
⎣-5 5 -2⎦

Step 2: Calculate A + (B + C)

A + (B + C) = ⎡4+10 3+6 1+8⎤
⎢2+2 3+7 -8+5⎥
⎣1-5 0+5 -4-2⎦

= ⎡14 9 9⎤
⎢4 10 -3⎥
⎣-4 5 -6⎦

Step 3: Calculate A + B

A + B = ⎡4+2 3+3 1+4⎤
⎢2+1 3+9 -8+2⎥
⎣1-7 0+1 -4-1⎦

= ⎡6 6 5⎤
⎢3 12 -6⎥
⎣-6 1 -5⎦

Step 4: Calculate (A + B) + C

(A + B) + C = ⎡6+8 6+3 5+4⎤
⎢3+1 12-2 -6+3⎥
⎣-6+2 1+4 -5-1⎦

= ⎡14 9 9⎤
⎢4 10 -3⎥
⎣-4 5 -6⎦

We can see that A + (B + C) = (A + B) + C! ✔️

Problem 3: Find X and Y

Given:

X + Y = ⎡7 0⎤
⎣3 5⎦

X - Y = ⎡3 0⎤
⎣0 4⎦

Find matrices X and Y.

Step 1: Add the two equations

(X + Y) + (X - Y) = 2X

⎡7 0⎤ + ⎡3 0⎤ = ⎡10 0⎤
⎣3 5⎦ ⎣0 4⎦ ⎣3 9⎦

So, 2X = ⎡10 0⎤
⎣3 9⎦

Step 2: Solve for X

X = ½ × ⎡10 0⎤
⎣3 9⎦

X = ⎡5 0⎤
⎣1.5 4.5⎦

Step 3: Subtract the equations

(X + Y) - (X - Y) = 2Y

⎡7 0⎤ - ⎡3 0⎤ = ⎡4 0⎤
⎣3 5⎦ ⎣0 4⎦ ⎣3 1⎦

So, 2Y = ⎡4 0⎤
⎣3 1⎦

Step 4: Solve for Y

Y = ½ × ⎡4 0⎤
⎣3 1⎦

Y = ⎡2 0⎤
⎣1.5 0.5⎦

Final Answer:

X = ⎡5 0⎤
⎣1.5 4.5⎦

Y = ⎡2 0⎤
⎣1.5 0.5⎦

Problem 4: Matrix Operations

Given matrices:

A = ⎡0 4 9⎤
⎣8 3 7⎦

B = ⎡7 3 8⎤
⎣1 4 9⎦

Find the value of:

(i) B - 5A

(ii) 3A - 9B

Part (i): B - 5A

First calculate 5A:

5A = ⎡0×5 4×5 9×5⎤
⎣8×5 3×5 7×5⎦ = ⎡0 20 45⎤
⎣40 15 35⎦

Now calculate B - 5A:

B - 5A = ⎡7-0 3-20 8-45⎤
⎣1-40 4-15 9-35⎦

= ⎡7 -17 -37⎤
⎣-39 -11 -26⎦

Part (ii): 3A - 9B

First calculate 3A and 9B separately:

3A = ⎡0 12 27⎤
⎣24 9 21⎦

9B = ⎡63 27 72⎤
⎣9 36 81⎦

Now calculate 3A - 9B:

3A - 9B = ⎡0-63 12-27 27-72⎤
⎣24-9 9-36 21-81⎦

= ⎡-63 -15 -45⎤
⎣15 -27 -60⎦

Problem 5: Find Values of Variables

(i) Find x, y, z if:

⎡x-3 3x-z⎤
⎣x+y+7 x+y+z⎦ = ⎡1 0⎤
⎣1 6⎦

(ii) Solve:

(x y-z z+3) + (y 4 3) = (4 8 16)

Part (i): Solving matrix equation

Set up equations from corresponding elements:

1. x - 3 = 1 ⇒ x = 4

2. 3x - z = 0 ⇒ 3(4) - z = 0 ⇒ z = 12

3. x + y + 7 = 1 ⇒ 4 + y + 7 = 1 ⇒ y = -10

4. x + y + z = 6 ⇒ 4 + (-10) + 12 = 6 ✔️

Solution: x = 4, y = -10, z = 12

Part (ii): Solving row matrix equation

Add the matrices and set up equations:

(x+y (y-z)+4 (z+3)+3) = (4 8 16)

1. x + y = 4

2. y - z + 4 = 8 ⇒ y - z = 4

3. z + 3 + 3 = 16 ⇒ z = 10

Substitute z = 10 into equation 2: y - 10 = 4 ⇒ y = 14

Substitute y = 14 into equation 1: x + 14 = 4 ⇒ x = -10

Solution: x = -10, y = 14, z = 10

Problem 6: Find x and y

Find x and y if:

x⎡4⎤
⎣-3⎦ + y⎡-2⎤
⎣3⎦ = ⎡4⎤
⎣6⎦

Step 1: Perform the scalar multiplication and addition

⎡4x - 2y⎤
⎣-3x + 3y⎦ = ⎡4⎤
⎣6⎦

Step 2: Set up the system of equations

1. 4x - 2y = 4

2. -3x + 3y = 6

Step 3: Simplify the equations

Equation 1 ÷ 2: 2x - y = 2 ⇒ y = 2x - 2

Equation 2 ÷ 3: -x + y = 2

Step 4: Substitute and solve

Substitute y = 2x - 2 into second equation:

-x + (2x - 2) = 2 ⇒ x - 2 = 2 ⇒ x = 4

Then y = 2(4) - 2 = 6

Solution: x = 4, y = 6

Problem 7: Solve Matrix Equation

Find non-zero values of x satisfying:

x⎡2x 2⎤
⎣3 x⎦ + 2⎡8 5x⎤
⎣4 4x⎦ = 2⎡x²+8 24⎤
⎣10 6x⎦

Step 1: Perform scalar multiplications

Left side becomes:

⎡2x² 2x⎤
⎣3x x²⎦ + ⎡16 10x⎤
⎣8 8x⎦ = ⎡2x²+16 2x+10x⎤
⎣3x+8 x²+8x⎦

Right side:

⎡2x²+16 48⎤
⎣20 12x⎦

Step 2: Set corresponding elements equal

1. 2x² + 16 = 2x² + 16 (always true)

2. 2x + 10x = 48 ⇒ 12x = 48 ⇒ x = 4

3. 3x + 8 = 20 ⇒ 3x = 12 ⇒ x = 4

4. x² + 8x = 12x ⇒ x² - 4x = 0 ⇒ x(x - 4) = 0 ⇒ x = 0 or 4

Step 3: Find common solution

From all equations, the only non-zero solution is x = 4

Solution: x = 4

Problem 8: Solve for x and y

⎡x²⎤
⎣y²⎦ + 2⎡-2x⎤
⎣-y⎦ = ⎡5⎤
⎣8⎦

Step 1: Perform the operations

⎡x² - 4x⎤
⎣y² - 2y⎦ = ⎡5⎤
⎣8⎦

Step 2: Set up equations

1. x² - 4x = 5 ⇒ x² - 4x - 5 = 0

2. y² - 2y = 8 ⇒ y² - 2y - 8 = 0

Step 3: Solve the quadratic equations

For x:

x² - 4x - 5 = 0 ⇒ (x - 5)(x + 1) = 0 ⇒ x = 5 or x = -1

For y:

y² - 2y - 8 = 0 ⇒ (y - 4)(y + 2) = 0 ⇒ y = 4 or y = -2

Solutions:

1. x = 5, y = 4

2. x = 5, y = -2

3. x = -1, y = 4

4. x = -1, y = -2

Practice What You Learned!

Let's test your understanding with some interactive problems:

Quick Quiz 1

If ⎡2 5⎤
⎣3 x⎦ + ⎡y 1⎤
⎣4 2⎦ = ⎡7 6⎤
⎣7 8⎦, find x and y

x =

y =

Quick Quiz 2

Solve: 3⎡x⎤
⎣y⎦ - ⎡4⎤
⎣2⎦ = ⎡5⎤
⎣7⎦

x =

y =