Find the value of a,b,c, and d from the equation: \(\begin{bmatrix}a-b&2a+c\\2a-b&3c+d\end{bmatrix}=\begin{bmatrix}-1&5\\0&13\end{bmatrix}\)
Find the value of a,b,c, and d from the equation: \(\begin{bmatrix}a-b&2a+c\\2a-b&3c+d\end{bmatrix}=\begin{bmatrix}-1&5\\0&13\end{bmatrix}\)
Solution and Explanation
\(\begin{bmatrix}a-b&2a+c\\2a-b&3c+d\end{bmatrix}=\begin{bmatrix}-1&5\\0&13\end{bmatrix}\)As the two matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
a − b = −1 … (1)
2a − b = 0 … (2)
2a + c = 5 … (3)
3c + d = 13 … (4)
From (2), we have:
b = 2a
Then, from (1), we have:
a − 2a = −1
\(\Rightarrow\) a = 1
\(\Rightarrow\)b = 2
Now, from (3), we have:
2 ×1 + c = 5
\(\Rightarrow\) c = 3
From (4) we have:
3 ×3 + d = 13
\(\Rightarrow\) 9 + d = 13 ⇒ d = 4
∴a = 1, b = 2, c = 3, and d = 4
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.