Matrix B Of Order 2×3: Find It Here!

Alright guys, let's dive into a cool math problem! We've got a matrix B, and it's a 2×3 matrix. That means it has 2 rows and 3 columns. What makes this interesting is that each element in the matrix is determined by a simple formula: $B_{ij} = 3i - 2j$. Our mission? Figure out what matrix B actually looks like. Let's break it down step by step so it’s super easy to follow!

Understanding the Matrix Order

First, let’s get cozy with what a 2×3 matrix means. Picture this: you have rows going horizontally and columns standing vertically. A 2×3 matrix will have 2 rows and 3 columns, making a grid of numbers. Each spot in this grid is an element, and we label them using subscripts. For example, $B_{11}$ is the element in the first row and first column, $B_{23}$ is the element in the second row and third column, and so on. Understanding this setup is crucial because it tells us exactly where each number we calculate will go. Now, why is this important? Because without understanding the structure, we'd be throwing numbers around without knowing where they belong, kind of like trying to assemble furniture without the instructions. So, take a moment to visualize this 2x3 grid in your mind. You've got two lines going across and three lines standing up, creating six little boxes where our numbers will live. Got it? Great! We're one step closer to cracking this matrix puzzle. Keep this image in mind as we move forward, because it's the foundation for everything else we're about to do.

The Formula: $B_{ij} = 3i - 2j$

Now, let's decode the formula that governs our matrix elements: $B_{ij} = 3i - 2j$. What this means is that to find the value of the element in the $i$-th row and $j$-th column, we plug $i$ and $j$ into this equation. For instance, to find $B_{11}$, we substitute $i = 1$ and $j = 1$ into the formula, which gives us $B_{11} = 3(1) - 2(1) = 1$. Easy peasy, right? This formula is like a magic recipe that tells us exactly what number to put in each spot of our matrix. The $i$ and $j$ are just placeholders, telling us which row and column we're currently working on. Think of it like a treasure map where $i$ and $j$ are the coordinates leading us to the correct value. So, when we see $B_{23}$, we know we need to find the treasure at the second row and third column. And the formula $3i - 2j$ is the key to unlocking that treasure. This formula is super important because it's the engine that drives our calculations. Without it, we'd just be guessing numbers. So, make sure you understand how to plug in the values of $i$ and $j$ to get the corresponding matrix element. Once you've got this down, the rest is just a matter of plugging and chugging. So, let's keep this formula in mind as we move forward and calculate each element of our matrix.

Calculating Each Element

Okay, let's roll up our sleeves and calculate each element of matrix B. We'll go through each position systematically to make sure we don't miss anything. Remember, our matrix is 2×3, so we have six elements to find: $B_{11}$, $B_{12}$, $B_{13}$, $B_{21}$, $B_{22}$, and $B_{23}$.

• $B_{11}$: For the first row and first column, $i = 1$ and $j = 1$. Plugging these into our formula, we get $B_{11} = 3(1) - 2(1) = 3 - 2 = 1$.
• $B_{12}$: For the first row and second column, $i = 1$ and $j = 2$. So, $B_{12} = 3(1) - 2(2) = 3 - 4 = -1$.
• $B_{13}$: For the first row and third column, $i = 1$ and $j = 3$. Thus, $B_{13} = 3(1) - 2(3) = 3 - 6 = -3$.
• $B_{21}$: For the second row and first column, $i = 2$ and $j = 1$. This gives us $B_{21} = 3(2) - 2(1) = 6 - 2 = 4$.
• $B_{22}$: For the second row and second column, $i = 2$ and $j = 2$. Therefore, $B_{22} = 3(2) - 2(2) = 6 - 4 = 2$.
• $B_{23}$: Finally, for the second row and third column, $i = 2$ and $j = 3$. So, $B_{23} = 3(2) - 2(3) = 6 - 6 = 0$.

See? It's just plugging in the numbers and doing the math. Each element is like solving a mini-equation, and once we've done all six, we'll have all the pieces of our matrix puzzle. This step-by-step approach is super helpful for avoiding mistakes and keeping track of where each number belongs. So, take your time, double-check your calculations, and make sure you're plugging the correct values of $i$ and $j$ into the formula. Once you've got all six elements, we're ready to assemble our final matrix and admire our handiwork. So, let's keep going, one element at a time, until we've conquered this matrix challenge.

Constructing the Matrix B

Now that we've calculated all the elements, let's put them together to form matrix B. Remember, matrix B is a 2×3 matrix, so it will look like this:

$B = \begin{bmatrix} B_{11} & B_{12} & B_{13} \ B_{21} & B_{22} & B_{23} \end{bmatrix}$

We found that:

• $B_{11} = 1$
• $B_{12} = -1$
• $B_{13} = -3$
• $B_{21} = 4$
• $B_{22} = 2$
• $B_{23} = 0$

Plugging these values into our matrix structure, we get:

$B = \begin{bmatrix} 1 & -1 & -3 \ 4 & 2 & 0 \end{bmatrix}$

And there you have it! That's our matrix B. We started with a formula and the dimensions of the matrix, and now we've successfully calculated each element and put them all together. This is a great feeling, right? You've taken something abstract and turned it into something concrete. You've conquered a matrix challenge! So, take a moment to appreciate your hard work and the problem-solving skills you've honed along the way. This is what math is all about: breaking down complex problems into smaller, manageable steps and then putting all the pieces back together to reveal the solution. So, congratulations on cracking this matrix puzzle. You've earned it!

Final Answer

So, to wrap it all up, the matrix B is:

$\begin{bmatrix} 1 & -1 & -3 \ 4 & 2 & 0 \end{bmatrix}$

Great job, guys! You've successfully determined matrix B. Keep up the awesome work, and remember, math is all about breaking things down and taking it one step at a time! You got this!