Matrix Operations

Idea. Two equal row vectors must match entry by entry. Turn each position into a small equation.
Step 1. First position. .
Step 2. Second position. .
So .

Idea. Multiplying by the identity leaves a matrix unchanged, just like multiplying a number by .
So and the product equals the second matrix itself.

Idea. Row–column multiplication: multiply matching entries and add. A times gives a number.

Think of as a plain number . When a scalar multiplies a matrix (or row), it multiplies each entry. So, That's all—entrywise scaling.

Row–column rule: Top entry . Bottom entry .

Right matrix is (twice the identity). Multiplying any matrix by doubles each column: First column ; second column . So product .

Multiply each entry by :

The given matrix is the identity . Identity is the multiplicative neutral element: Therefore,

We are given the matrix: To multiply a scalar (5) with a matrix, simply multiply each element of the matrix by 5: Thus, the result is .

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second matrix. Here, the second matrix is the identity matrix: Multiplying any matrix with the identity matrix will return the original matrix. Therefore, multiplying: Thus, the correct answer is option (B).

We are given two matrices and are asked to multiply them: This multiplication involves calculating the dot product of the row vector and the column vector. The result will be a scalar: Thus, the result is:

We are asked to multiply two matrices: Performing the multiplication: - First row, first column: - First row, second column: - Second row, first column: - Second row, second column: Thus, the resulting matrix is:

The given matrix multiplication involves multiplying: The product will result in a single element, since we are multiplying a 2x2 matrix with a 2x1 matrix: Thus, the resulting matrix is:

We are given the matrix multiplication: We perform the matrix multiplication to find the result. The product of a 2x2 matrix by another 2x2 matrix is calculated by taking the sum of the products of corresponding elements from each row and column: - - Thus, the resulting matrix is: (D) is the correct answer.

We are multiplying two matrices: Matrix multiplication involves multiplying corresponding elements and summing them. The result is a 1x1 matrix: Therefore, the correct answer is .

We are given two matrices and we are asked to multiply them: Matrix multiplication follows the rule that the element in the -th row and -th column of the resulting matrix is the sum of the products of corresponding elements from the rows of the first matrix and columns of the second matrix. Performing matrix multiplication, we get the result:

We are given two matrices and are asked to multiply them: We multiply these matrices by computing the dot product of rows from the first matrix with columns from the second matrix. For the first element of the product matrix: For the second element: For the third element: For the fourth element: Thus, the final product matrix is: