If $\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$, then the value of $a + 2b - 3c + 4d$ is equal to

When two matrices are equal, each corresponding position has the same value.

$\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$

Matching each position:

$2a + b = 4$ ... equation (1)

$a - 2b = -3$ ... equation (2)

$5c - d = 11$ ... equation (3)

$4c + 3d = 24$ ... equation (4)

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From equation (2):

$a - 2b = -3$

$a = 2b - 3$ ... (5)

Substituting into equation (1):

$2a + b = 4$

$2(2b - 3) + b = 4$

$4b - 6 + b = 4$

$5b = 10$

$b = 2$

Using equation (5):

$a = 2(2) - 3$

$a = 4 - 3$

$a = 1$

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From equation (3):

$5c - d = 11$

$d = 5c - 11$ ... (6)

Substituting into equation (4):

$4c + 3d = 24$

$4c + 3(5c - 11) = 24$

$4c + 15c - 33 = 24$

$19c = 57$

$c = 3$

Using equation (6):

$d = 5(3) - 11$

$d = 15 - 11$

$d = 4$

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The expression $a + 2b - 3c + 4d$ becomes:

$a + 2b - 3c + 4d = 1 + 2(2) - 3(3) + 4(4)$

$= 1 + 4 - 9 + 16$

$= 12$

Therefore, the value of $a + 2b - 3c + 4d = 12$