Problem 66
Question
Find the sum or the difference of the matrices. $$ \left[\begin{array}{rrr}9 & 1 & 6 \\\\-4 & -7 & 1 \\\\-5 & 0 & -1\end{array}\right]+\left[\begin{array}{rrr}-6 & 3 & -5 \\\\-2 & 4 & -4 \\ 0 & 5 & 1\end{array}\right] $$
Step-by-Step Solution
Verified Answer
The resultant matrix of this operation is \[ \left[\begin{array}{ccc}3 & 4 & 1 \\-6 & -3 & -3 \\-5 & 5 & 0\end{array}\right] \]
1Step 1 Identify Matrices
Before proceeding with the operation, identify matrices involved in the operation. Here, two 3x3 matrices are being added. The first matrix \[ \left[\begin{array}{ccc}9 & 1 & 6 \-4 & -7 & 1 \-5 & 0 & -1 \end{array}\right]\] and the second matrix \[\left[\begin{array}{ccc}-6 & 3 & -5 \-2 & 4 & -4 \ 0 & 5 & 1\end{array}\right]\]
2Step 2 Addition of Matrices
Perform matrix addition by adding each corresponding element in both matrices. The elements are added as per their position in both matrices. In other words, the element in the first row and first column gets added to the element in the first row and first column of the second matrix, and so on.
3Step 3 Compute Resultant Matrix
Compute the resulting matrix from the operation. So, the resultant matrix will be: \[\left[\begin{array}{ccc}(9-6) & (1+3) & (6-5) \(-4-2) & (-7+4) & (1+(-4)) \(-5+0) & (0+5) & (-1+1) \end{array}\right]=\left[\begin{array}{ccc}3 & 4 & 1 \-6 & -3 & -3 \ -5 & 5 & 0\end{array}\right]\]
Key Concepts
MatricesElementary Operations3x3 MatrixAlgebraic Operations on Matrices
Matrices
A matrix is an ordered rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. For instance, a 3x3 matrix has three rows and three columns, and it contains nine elements in total. Matrices are widely used in various fields, including mathematics, physics, computer science, and engineering, for solving systems of linear equations, representing graphs and networks, and in algorithms.
Matrices are often denoted by uppercase letters and can be added, subtracted, and multiplied under specific rules. The concept of a matrix is central to linear algebra, and it's a powerful tool because it can encapsulate complex information in a compact form, which can then be manipulated to find solutions to a problem.
Matrices are often denoted by uppercase letters and can be added, subtracted, and multiplied under specific rules. The concept of a matrix is central to linear algebra, and it's a powerful tool because it can encapsulate complex information in a compact form, which can then be manipulated to find solutions to a problem.
Elementary Operations
Elementary operations are the basic manipulations that can be performed on matrices. These include the addition, subtraction, and multiplication of matrices by scalars, as well as matrix multiplication. However, not all operations are defined for all sizes of matrices; for instance, matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix.
When adding or subtracting matrices, they must be of the same dimension, meaning they have the same number of rows and columns. The elementary operations on matrices play a crucial role in various calculations, such as solving systems of equations using matrices, determining inverses of matrices, and more.
When adding or subtracting matrices, they must be of the same dimension, meaning they have the same number of rows and columns. The elementary operations on matrices play a crucial role in various calculations, such as solving systems of equations using matrices, determining inverses of matrices, and more.
3x3 Matrix
A 3x3 matrix is a special case of a square matrix, where there are three rows and three columns. The general form of a 3x3 matrix is
\[\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33} \end{array}\right]\]
where \(a_{ij}\) represents the element in the \(i\)th row and the \(j\)th column. Square matrices, including 3x3 matrices, have certain unique properties, such as the determinant and the possibility of finding the inverse (provided that the determinant is not zero). They are often used in computer graphics transformations and in the representation of more complex systems in physics.
\[\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33} \end{array}\right]\]
where \(a_{ij}\) represents the element in the \(i\)th row and the \(j\)th column. Square matrices, including 3x3 matrices, have certain unique properties, such as the determinant and the possibility of finding the inverse (provided that the determinant is not zero). They are often used in computer graphics transformations and in the representation of more complex systems in physics.
Algebraic Operations on Matrices
Algebraic operations on matrices include addition, subtraction, and multiplication. For matrix addition and subtraction, these operations are performed element by element. This means that the resulting matrix has the same dimensions as the original matrices, and each element of the resulting matrix is the sum or difference of the corresponding elements of the original matrices.
For our exercise, we are focusing on matrix addition. When adding two matrices, such as two 3x3 matrices, you simply take the corresponding elements from each matrix and add them together. For example, if you have elements \(a_{ij}\) from the first matrix and elements \(b_{ij}\) from the second matrix, the resulting matrix will have elements \(a_{ij} + b_{ij}\) at each position \(ij\). It is important to perform these operations carefully to ensure that each element is accurately computed.
For our exercise, we are focusing on matrix addition. When adding two matrices, such as two 3x3 matrices, you simply take the corresponding elements from each matrix and add them together. For example, if you have elements \(a_{ij}\) from the first matrix and elements \(b_{ij}\) from the second matrix, the resulting matrix will have elements \(a_{ij} + b_{ij}\) at each position \(ij\). It is important to perform these operations carefully to ensure that each element is accurately computed.
- The addition must be done element-wise.
- The matrices must be of the same size.
- The result will also be a matrix of the same size.
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