Problem 52 - Algebra 2

Question

Perform the indicated operations. $$ \left[\begin{array}{rr}{3} & {-4} \\ {2} & {8} \\ {0} & {1}\end{array}\right]+\left[\begin{array}{rr}{-5} & {0} \\ {7} & {7} \\ {3} & {-6}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{array}{rr}-2 & -4 \\ 9 & 15 \\ 3 & -5\end{array}\).

Step 1: Setup the Matrices for Addition

First, observe that the matrices given for addition are both 3x2 matrices, which means they have the same number of rows and columns. Being the same size allows us to add them element-wise.

Step 2: Add the Elements of the First Row

The first row of the first matrix is \([3, -4]\) and the first row of the second matrix is \([-5, 0]\). Calculate the sum of corresponding elements: - First element: \(3 + (-5) = -2\) - Second element: \(-4 + 0 = -4\)The sum for the first row is \([-2, -4]\).

Step 3: Add the Elements of the Second Row

The second row of the first matrix is \([2, 8]\) and the second row of the second matrix is \([7, 7]\). Calculate the sum of corresponding elements: - First element: \(2 + 7 = 9\) - Second element: \(8 + 7 = 15\)The sum for the second row is \([9, 15]\).

Step 4: Add the Elements of the Third Row

The third row of the first matrix is \([0, 1]\) and the third row of the second matrix is \([3, -6]\). Calculate the sum of corresponding elements: - First element: \(0 + 3 = 3\) - Second element: \(1 + (-6) = -5\)The sum for the third row is \([3, -5]\).

Step 5: Write Down the Resulting Matrix

Combine the sums of each row to form the new matrix, which is the result of adding the two matrices:\[\begin{array}{rr}{-2} & {-4} \{9} & {15} \{3} & {-5}\end{array}\]

Key Concepts

3x2 Matrices

Thinking about matrices for the first time? No worries, let's break them down. A 3x2 matrix is simply a rectangular array with 3 rows and 2 columns. It looks like this:

• First row has two elements
• Second row has two elements
• Third row has two elements

Each number in a matrix is known as an element. The position of each element is crucial, as it determines how we perform operations like addition. For a matrix to be classified as 3x2, it must indeed have the "3" for rows and "2" for columns. You can visualize it as three horizontal lines, each holding two numbers. When working with matrices, ensure they are of the same size to proceed with operations like addition smoothly.

Element-wise Operations

Imagine dealing with matrices as piquing each element's interest individually. When we add matrices, we do it element-wise. This means each element from one matrix pairs with its counterpart from another matrix. Here's how it works:

• Line up both matrices
• Add each pair of corresponding elements

For example, take the matrices:\(\begin{array}{cc}3 & -4 \2 & 8 \0 & 1\end{array}\)and \(\begin{array}{cc}-5 & 0 \7 & 7 \3 & -6\end{array}\)You add the top left element of the first matrix to the top left of the second matrix, then proceed element by element, row by row. This operation ensures simplicity and accuracy by keeping the structure intact. Always remember, element-wise operations meet in corresponding spots of the matrix.

Matrix Arithmetic

Matrix arithmetic extends beyond simple numbers; it's a structured form of calculations with arrays. A fundamental operation in matrix arithmetic is addition. To add two matrices, it's crucial that both matrices are of the same dimensions. Once confirmed, proceed by adding corresponding elements from each matrix, one by one. At its core, matrix addition involves:

• Ensuring matrices share the same dimensionality
• Adding each element in one matrix to the corresponding element in the other
• Rebuilding the newly formed matrix from these sums

Knowing this, if you were faced with adding two matrices like:\(\begin{array}{cc}3 & -4 \2 & 8 \0 & 1\end{array}\) and \(\begin{array}{cc}-5 & 0 \7 & 7 \3 & -6\end{array}\), you calculate all corresponding sums and form a new matrix. It becomes a practical and systematic approach to matrix calculations.