If A = [[1, 2, 2], [2, 1, -2], [a, 2, b]] is a matrix satisfying A * A^T = 9*I, find the values of a and b.

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Step-by-Step Solution

Step 1: Calculate the transpose of A

To find the transpose of matrix A$, we interchange its rows and columns. The first row of A becomes the first column of A^T$, the second row becomes the second column, and so on.

Step 2: Calculate the product A * A^T

Next, we multiply matrix A by its transpose A^T$. Each element of the resulting matrix is obtained by taking the dot product of a row from A and a column from A^T$. We perform the necessary multiplications and additions.

Step 3: Calculate 9I

The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere. Multiplying it by a scalar 9 means multiplying each element of the identity matrix by 9$.

Step 4: Equate A * A^T with 9I and solve for a and b

By equating the corresponding elements of the two matrices A A^T and 9I$, we obtain a system of three equations. We will use the first two equations to solve for a and b$.

Step 5: Solve the system of linear equations

We have a system of two linear equations with two variables. We can solve this system by adding the two equations to eliminate b$, which allows us to find a$. Then, we substitute the value of a back into one of the equations to find b$.

Step 6: Verify the values using the third equation

Finally, we substitute the calculated values of a = -2 and b = -1 into the third equation a^2+b^2+4 = 9 to verify that they satisfy all conditions. Since 9=9$, our values are correct.