Example Question - solving matrix equation
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Solving Matrix Equation with Variable
The image you’ve provided shows a matrix equation that needs to be solved. The quality of the image is quite blurry and the text is not very clear, but I can decipher that the equation is of the form: \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 \\ x \end{bmatrix} = \begin{bmatrix} e & f & g \\ h & i & j \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ -x \end{bmatrix} \] Where \( a, b, c, d, e, f, g, h, i, j \) are numbers and \( x \) is the variable we need to solve for. However, I'm unable to precisely read the numbers due to the lack of clarity. To solve the matrix equation, one would typically multiply out both sides to obtain two systems of linear equations, and then solve for \( x \). Since I cannot provide an accurate answer without the specific values, I can only guide you with the general approach: 1. Multiply the left-hand side matrix by the column matrix with variables \( 1 \) and \( x \). 2. Multiply the right-hand side matrix by the column matrix with variables \( 1 \), \( 0 \), and \( -x \). 3. Equate the corresponding elements of the resulting matrices from both sides. 4. Solve the system of equations to obtain the value of \( x \). If you can provide a clearer image or the specific values of the numbers, I could proceed with the calculation and give you a more precise answer.
Solving a Matrix Equation for x and y
You want to solve the matrix equation to find the values of x and y. The equation is given as: \[ \begin{pmatrix} 4 & 2 \\ -2 & x \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} y & 10 \\ -2 & 4 \end{pmatrix} \] To find the values of x and y, you need to multiply the first two matrices and set the resulting matrix equal to the one with y and x: First, compute the product of the two matrices on the left side: \[ \begin{pmatrix} 4 & 2 \\ -2 & x \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4(1) + 2(0) & 4(2) + 2(1) \\ -2(1) + x(0) & -2(2) + x(1) \end{pmatrix} \] \[ = \begin{pmatrix} 4 & 8 + 2 \\ -2 & -4 + x \end{pmatrix} \] \[ = \begin{pmatrix} 4 & 10 \\ -2 & x - 4 \end{pmatrix} \] Now, equate this resulting matrix to the matrix on the right side of the equation and solve for x and y: \[ \begin{pmatrix} 4 & 10 \\ -2 & x - 4 \end{pmatrix} = \begin{pmatrix} y & 10 \\ -2 & 4 \end{pmatrix} \] For the matrices to be equal, their corresponding elements must be equal, so: \[ y = 4 \quad \text{and} \quad x - 4 = 4 \] From the second equation: \[ x = 4 + 4 \] \[ x = 8 \] Therefore, the values of x and y are 8 and 4, respectively.
Solving Matrix Equation for Variables
The image shows a matrix equation that we need to solve to find the values of p and q. The matrix equation is given as: ``` [ 4/p p ] [ 7(q) 11(p) ] [-1/q -p ] = [ -3 13 ] ``` To solve for p and q, we need to equate elements in the corresponding positions from the given two matrices. That will give us a system of equations as follows: 1. (4/p) * 7(q) + p * (-3) = 11(p) 2. (-1/q) * 7(q) + (-p) * (-3) = 13 Let's solve these equations. For the first equation: (4q/p) * 7 + p * (-3) = 11p 28q/p - 3p = 11p 28q = 14p^2 q = (14p^2)/28 q = p^2/2 (1) For the second equation: ((-7) + 3p) = 13 3p - 7 = 13 3p = 20 p = 20/3 Now we can substitute the value of p into equation (1) to find q: q = (20/3)^2/2 q = 400/9 * 1/2 q = 400/18 q = 200/9 q = 22.222... Therefore, the values of p and q are: p = 20/3 q = 200/9 However, note that the way both equations are constructed implies that there is an element of matrix multiplication involved, which is not correctly represented in the image. Normally, when multiplying two matrices, the elements are multiplied and then added together according to matrix multiplication rules. The image seems to imply direct multiplication which would defy the matrix multiplication rules. But based on the provided equations and assuming they are correctly representing a matrix product, the above solution for p and q is correct.
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