Let's learn about the determinant of a 3x3 matrix! 🎉
1️⃣ Imagine you have a box with numbers arranged like this:
**Box 2:** 🍎 🍎 🍎
**Box 3:** 🍎 🍎 🍎
Multiply groups of apples diagonally, then subtract other groups diagonally. 4️⃣ Now do the math: $$1 \times 5 \times 9 = 45$$ $$2 \times 6 \times 7 = 84$$ $$3 \times 4 \times 8 = 96$$ Add these: $$45 + 84 + 96 = 225$$ Then: $$3 \times 5 \times 7 = 105$$ $$1 \times 6 \times 8 = 48$$ $$2 \times 4 \times 9 = 72$$ Add these: $$105 + 48 + 72 = 225$$ 5️⃣ Finally, subtract: $$225 - 225 = 0$$ So, the determinant is 0! Great job! 🎯 The determinant helps us understand if the matrix can be reversed or not. Want to try with your own numbers? Just ask!
Matrix:
1️⃣ 2️⃣ 3️⃣
4️⃣ 5️⃣ 6️⃣
7️⃣ 8️⃣ 9️⃣
(3 rows and 3 columns)
2️⃣ To find the determinant, we do a special math trick:
We multiply and add numbers in groups:
- First, multiply diagonals from top-left to bottom-right:
$$1 \times 5 \times 9 + 2 \times 6 \times 7 + 3 \times 4 \times 8$$
- Then subtract the diagonals from top-right to bottom-left:
$$3 \times 5 \times 7 + 1 \times 6 \times 8 + 2 \times 4 \times 9$$
3️⃣ Let's see it with apples 🍎 (imagine apples in 3 rows and 3 columns):
**Box 1:**
🍎 🍎 🍎1️⃣ 2️⃣ 3️⃣
4️⃣ 5️⃣ 6️⃣
7️⃣ 8️⃣ 9️⃣
(3 rows and 3 columns)
**Box 2:** 🍎 🍎 🍎
**Box 3:** 🍎 🍎 🍎
Multiply groups of apples diagonally, then subtract other groups diagonally. 4️⃣ Now do the math: $$1 \times 5 \times 9 = 45$$ $$2 \times 6 \times 7 = 84$$ $$3 \times 4 \times 8 = 96$$ Add these: $$45 + 84 + 96 = 225$$ Then: $$3 \times 5 \times 7 = 105$$ $$1 \times 6 \times 8 = 48$$ $$2 \times 4 \times 9 = 72$$ Add these: $$105 + 48 + 72 = 225$$ 5️⃣ Finally, subtract: $$225 - 225 = 0$$ So, the determinant is 0! Great job! 🎯 The determinant helps us understand if the matrix can be reversed or not. Want to try with your own numbers? Just ask!