If A = [[1, 2], [3, 4]], find A^2.

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Question: If A = [[1, 2], [3, 4]], find A^2.

Options:

Correct Answer: [7, 10], [15, 22]

Solution:

A^2 = A * A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].

If A = [[1, 2], [3, 4]], find A^2.

If A = [[1, 2], [3, 4]], find A^2.

Correct Answer: [[7, 10], [15, 22]]

Step 1: Identify the matrix A, which is A = [[1, 2], [3, 4]].
Step 2: To find A^2, we need to multiply A by itself: A^2 = A * A.
Step 3: Set up the multiplication of the two matrices. The result will also be a 2x2 matrix.
Step 4: Calculate the first element of the resulting matrix: (1*1 + 2*3). This equals 1 + 6 = 7.
Step 5: Calculate the second element of the first row: (1*2 + 2*4). This equals 2 + 8 = 10.
Step 6: Calculate the first element of the second row: (3*1 + 4*3). This equals 3 + 12 = 15.
Step 7: Calculate the second element of the second row: (3*2 + 4*4). This equals 6 + 16 = 22.
Step 8: Combine all the calculated elements to form the resulting matrix: A^2 = [[7, 10], [15, 22]].

Matrix Multiplication – Understanding how to multiply two matrices, including the rules for dimensions and the calculation of each element.
Matrix Squaring – Applying matrix multiplication to find the square of a matrix, which involves multiplying the matrix by itself.