Question 9

Explanation

Given the matrix equation:

Where $A = \begin{pmatrix} 6 & 3 \\ b & 14 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 2 \\ 4 & 5 \end{pmatrix}$.

If both sides are multiplied by $M^{-1}$ from the right, then:

Substitute the known matrices:

Perform matrix multiplication on the right side:

From the equality of two matrices, we obtain:

Element at row 1 column 1:

Element at row 2 column 1:

Now we check the truth of each statement.

Statement 1

States that $a = 2$. Since the calculated value is $a = 0$, this statement is False.

Statement 2

States that $a + b = 15$. Calculate $a + b$:

This statement is True.

Statement 3

States that the determinant of the inverse matrix $M$ is $3$. The inverse matrix $M$ is $M^{-1} = \begin{pmatrix} a & 1 \\ 3 & 2 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 3 & 2 \end{pmatrix}$.

The determinant of $M^{-1}$ is:

Since the determinant result is $-3$, not $3$, this statement is False.

Conclusion

The correct sequence of truth values is False, True, False.