problem 3. use determinant to find out whether the following matrix a is invertible. 5 11 a-11-3-2
Use the determinant to find out whether the following matrix ( A ) is invertible: (\begin{pmatrix} 5 & 11 & a \ -1 & -3 & -2 \end{pmatrix})
Answer:
To determine whether a matrix is invertible, we need to check if its determinant is zero. If the determinant is zero, the matrix is not invertible (singular); if the determinant is non-zero, the matrix is invertible (non-singular).
However, there seems to be a misunderstanding in the given problem statement. The matrix ( A ) is presented as:
A = \begin{pmatrix} 5 & 11 & a \\ -1 & -3 & -2 \end{pmatrix}
This is a ( 2 \times 3 ) matrix, and only square matrices (( n \times n )) have determinants. For invertibility, we generally consider square matrices.
Assuming the problem should involve a ( 3 \times 3 ) matrix, let’s complete the matrix by hypothesizing that the matrix might be:
A = \begin{pmatrix} 5 & 11 & a \\ -1 & -3 & -2 \\ c_1 & c_2 & c_3 \end{pmatrix}
However, without knowing the values of ( c_1, c_2, ) and ( c_3 ), we cannot proceed. Therefore, we should focus on the given ( 2 \times 2 ) matrix:
B = \begin{pmatrix} 5 & 11 \\ -1 & -3 \end{pmatrix}
We will determine if this ( 2 \times 2 ) part is invertible:
Step 1: Calculate the determinant
For a ( 2 \times 2 ) matrix:
B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
the determinant is given by:
\text{det}(B) = ad - bc
For our specific matrix:
\text{det}(B) = 5 \times -3 - 11 \times -1
Calculating the Determinant:
\text{det}(B) = (5 \cdot -3) - (11 \cdot -1)
\text{det}(B) = -15 + 11
\text{det}(B) = -4
Since the determinant is ( -4 ), which is non-zero, the matrix ( B ) is invertible.
Conclusion:
Based on the ( 2 \times 2 ) segment of the initial specified matrix, we conclude that this part is invertible. However, since only the ( 2 \times 2 ) matrix can be inverted and the original matrix was not properly formatted for invertibility in matrix algebra (since it was ( 2 \times 3 )), it is crucial to revise the problem statement to a proper ( n \times n ) form in future tasks.
Therefore, the 2 \times 2 matrix:
B = \begin{pmatrix} 5 & 11 \\ -1 & -3 \end{pmatrix}
is invertible. To confirm invertibility for a square 3 \times 3 matrix, a complete matrix must be provided.