If A = [[1, 2, 2], [2, 1, -2], [a, 2, b]] satisfies A*A^T = 9*I_3, find the values of a and b and compute det(adj(A)).
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Step-by-Step Solution
Step 1: Calculate the transpose of A
First, we need to find the transpose of matrix $A$. The transpose of a matrix is obtained by interchanging its rows and columns. So, the first row of $A$ becomes the first column of $A^T$, the second row becomes the second column, and so on.
Step 2: Compute 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 matrix multiplication carefully, summing the products of corresponding elements.
Step 3: Equate A A^T to 9 I_3 and solve for a and b
We are given that $A A^T = 9 I_3$, where $I_3$ is the $3 \times 3$ identity matrix. We equate the elements of the computed $A A^T$ with the elements of $9 I_3$. This gives us a system of three equations with two unknowns, $a$ and $b.
Step 4: Solve the system of equations for a and b
Now we solve the system of linear equations for $a$ and $b$. Adding equations (1) and (2) eliminates $b$, allowing us to solve for $a$. Then, we substitute the value of $a$ back into equation (1) to find $b$. Finally, we verify these values using equation (3).
Step 5: Compute det(A)
With the values of $a = -2$ and $b = -1$, we can write the complete matrix $A$. Now, we calculate the determinant of $A$. For a $3 \times 3$ matrix, we use the cofactor expansion method along the first row.
Step 6: Compute det(adj(A))
Finally, we need to compute the determinant of the adjoint of $A$. We use the property that for an $n \times n$ matrix $A$, $\det(\text{adj}(A)) = (\\det(A))^{n-1}$. Since $A$ is a $3 \times 3$ matrix, $n=3$. We substitute the calculated value of $\\det(A)$ into this formula.