Problem 102
Question
Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $$C(A+2 B)$$
Step-by-Step Solution
Verified Answer
Yes, the operation \(C(A+2B)\) is possible and the resulting matrix will have dimensions \(3 \times 3\).
1Step 1: Check Dimensions for Matrix Addition
You start with the inside of the parentheses: \(A+2B\). The matrices \(A\) and \(B\) have the same dimensions, \(2 \times 3\), and therefore can be added together. The scalar multiplication \(2B\) won't change the dimensions of \(B\). Therefore, \(A + 2B\) is possible and has dimensions \(2 \times 3\).
2Step 2: Check Dimensions for Matrix Multiplication
The next step is to check whether \(C(A+2B)\) is valid. For matrix multiplication, the number of columns in the first matrix must equal to the number of rows in the second matrix. Matrix \(C\) has dimensions \(3 \times 2\) and the result of \(A+2B\) has dimensions \(2 \times 3\). Therefore, their multiplication is possible.
3Step 3: Dimension of the Resulting Matrix
Finally, the dimension of the resulting matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix. So, for this case, the multiplication of \(C(A+2B)\) will give a matrix with dimensions \(3 \times 3\).
Key Concepts
Matrix DimensionsMatrix AdditionScalar Multiplication
Matrix Dimensions
Understanding matrix dimensions is crucial when performing operations such as addition, multiplication, or scalar multiplication. Matrix dimensions are expressed as 'rows \times columns', and they indicate the size of a matrix. For instance, a matrix with dimensions of \(2 \times 3\) has two rows and three columns.
When analyzing whether operations can be performed on two or more matrices, their dimensions play a key role. For example, in the operation \(C(A+2B)\), we need to consider the dimensions of each matrix to determine if the operation is feasible. If you encounter a pair of matrices with the same number of rows and columns, like matrices \(A\) and \(B\) both being \(2 \times 3\), you can proceed with operations such as addition or scalar multiplication. However, if their dimensions do not match, these operations cannot be performed.
When you are multiplying two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. As a quick tip for remembering, you can think of an 'inner dimension rule': the inner dimensions of the matrices being multiplied must match for the operation to be valid. For instance, if you're multiplying a \(3 \times 2\) matrix with a \(2 \times 3\) matrix, the multiplication is valid, and the resulting matrix will have dimensions defined by the outer numbers, resulting in a \(3 \times 3\) matrix.
When analyzing whether operations can be performed on two or more matrices, their dimensions play a key role. For example, in the operation \(C(A+2B)\), we need to consider the dimensions of each matrix to determine if the operation is feasible. If you encounter a pair of matrices with the same number of rows and columns, like matrices \(A\) and \(B\) both being \(2 \times 3\), you can proceed with operations such as addition or scalar multiplication. However, if their dimensions do not match, these operations cannot be performed.
When you are multiplying two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. As a quick tip for remembering, you can think of an 'inner dimension rule': the inner dimensions of the matrices being multiplied must match for the operation to be valid. For instance, if you're multiplying a \(3 \times 2\) matrix with a \(2 \times 3\) matrix, the multiplication is valid, and the resulting matrix will have dimensions defined by the outer numbers, resulting in a \(3 \times 3\) matrix.
Matrix Addition
When it comes to matrix addition, the rule is straightforward: you can only add matrices if they have the exact same dimensions. That is, if two matrices share the same number of rows and the same number of columns, they can be added element-wise.
In the case of our matrices \(A\) and \(B\), both have dimensions of \(2 \times 3\), which simplifies the process. You simply add corresponding entries of these two matrices, meaning you add the top-left entries of \(A\) and \(B\) together, then the top-middle entries together, and so on, until all positions have been summed. This also means that the resulting matrix will maintain the same dimensions, \(2 \times 3\), as the original matrices.
Remember that matrix addition is also commutative, so \(A+B\) will give you the same result as \(B+A\). Moreover, if you're asked to add a matrix to itself as in \(A+A\), you're effectively doubling each element of matrix \(A\), which is a neat connection to the concept of scalar multiplication.
In the case of our matrices \(A\) and \(B\), both have dimensions of \(2 \times 3\), which simplifies the process. You simply add corresponding entries of these two matrices, meaning you add the top-left entries of \(A\) and \(B\) together, then the top-middle entries together, and so on, until all positions have been summed. This also means that the resulting matrix will maintain the same dimensions, \(2 \times 3\), as the original matrices.
Remember that matrix addition is also commutative, so \(A+B\) will give you the same result as \(B+A\). Moreover, if you're asked to add a matrix to itself as in \(A+A\), you're effectively doubling each element of matrix \(A\), which is a neat connection to the concept of scalar multiplication.
Scalar Multiplication
Moving on to scalar multiplication, this operation involves multiplying a matrix by a scalar, which is just a fancy term for a regular number. When you multiply a matrix by a scalar, you multiply each entry of the matrix by that number.
Let's consider the scalar multiplication of \(2B\) from our exercise. Here, the scalar is 2, and matrix \(B\) has dimensions of \(2 \times 3\). As you would expect, every element of matrix \(B\) is multiplied by 2. The beauty of scalar multiplication lies in its simplicity. Regardless of how large or small the scalar is, the dimensions of the matrix remain unchanged. So, \(2B\) still has a dimension of \(2 \times 3\), just like matrix \(B\) before the multiplication.
This is a useful property because it allows us to easily combine scalar multiplication with other operations like matrix addition or matrix multiplication, without worrying about altering the dimensionality of our matrices during the intermediate steps.
Let's consider the scalar multiplication of \(2B\) from our exercise. Here, the scalar is 2, and matrix \(B\) has dimensions of \(2 \times 3\). As you would expect, every element of matrix \(B\) is multiplied by 2. The beauty of scalar multiplication lies in its simplicity. Regardless of how large or small the scalar is, the dimensions of the matrix remain unchanged. So, \(2B\) still has a dimension of \(2 \times 3\), just like matrix \(B\) before the multiplication.
This is a useful property because it allows us to easily combine scalar multiplication with other operations like matrix addition or matrix multiplication, without worrying about altering the dimensionality of our matrices during the intermediate steps.
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